Edit: Because I got into the weeds a bit, to answer the question, in the United States, Impeachment is not a legal matter but a political matter (which is what the DOJ handles). In fact, the Supreme Court of the United States has held that Impeachment is not subject to judicial review because it's not a right of the judiciary, but the legislature. The DOJ declining to investigate anyone with respect to the phone call only means that no one will go to jail. Trump may be Impeached over the matter, which if upheld in the Senate, will result in only two punishments: 1.) Immediate removal from Office. 2.) Possible loss of the right to hold Federal Office (Impeached people who are removed are still able to serve in state elected offices provided those states haven't also impeached him... though they can't impeach for anything done in Federal Office). The second punsihment is not a garentee and at least one person Impeached by Congress (US Represetative (Then Federal Judge) Alcee Hastings was impeached and removed from office in 1989, but never barred from serving in Federal Office.). It should also be mentioned that while only members of the Judicial and Executive Branch can be impeached, the second punishment issued bars service in any office, in any branch, either elected or appointed so those so barred cannot serve in Congress. Impeachment does not immunize the impeached person from legal action related to the offenses, as it is not a legal criminal matter, and thus double jeopardy does not apply.
So one of the big things to yesterdays news cycle to be understood is that Speaker of the House Nancy Pelosi (the highest democrat in elected office at time of writing), has flipped on her position of impeachment investigation. This does not mean an impeachment is going to happen... just that they are now looking into the possibility of an impeachable offense.
Prior to Tuesday Afternoon, Nancy Pelosi was stratigically opposed to impeachment of Trump, and was the strongest oppostion from within her own party. This is because Impeachment has been somewhat unpopular with Moderate democratic voters and more importantly, very unpopular with unaffiliated/independent voters. To keep a majority in the House and possibly take the senate, Pelosi needs to keep both of these factions happy (she is very unpopular with the hard line left and hard line right. Moderate right Americans generally tolerate her right now under the view that if it's between her and a far left speaker, they'd prefer her. Moderate Left support her (though I know some who are saying better the devil you know)). The unaffiliated/independents are actually very important in the United States as they are rather sizable block of the general election voter. While the U.S. has only two viable parties, these parties are "big tent" compared to multi-party systems which can have a narrow focus and form a coalition with other parties. For example, the Democrats generally have the same opinions on Green Politics as the U.S. Green Party, so people who vote on Climate issues first tend to vote Democrat because they can win. The trade off is that the parties in the United States don't have much control over their elected membership, and they can vote against the party interest (often because the constituent's interests are not aligned with the party interest. If your in a swing seat, you better hope the constituents think you're voting with them.).
There are three types of these voters, two of which are basically of the same mold of thinking, just painted a different color: A moderate who has a preferred party but is not committed either because they know their party isn't without sin or because they don't support the hardliners of that party. They'll probably vote against your big rival... but that doesn't mean they'll support you if your terrible. The other flavor is more of a true neutral. They aren't going to vote for either party, but they will vote for the best man for the job (AKA "the lesser of two evils." Americans generally trust politicians to be untrustworthy. Cockroaches enjoy a better approval rating than Congress among Americans. Both steal from you and are sick filthy creatures, but Cockroaches don't lie to you. The mentality is well known as there are always bumper stickers for voting for popular fictional villains because why be content with a Lesser Evil.).
What makes the unaffiliated/independent voter so important is that they are actually a large segment of the voting population and registering as such has been on a 50+ year increasing trend and are close to being the majority of political party affiliation of registered voters if they haven't eclipsed both parties already. They are dismissed at a candidate's peril. Democrats and Republicans may start the fight, but the unaffiliated/independent block decides who wins. It isn't helped that there are many reasons to register as one, but the general unifying idea is that both parties aren't deserving of the vote just because their terrible candidate is not as bad as the best candidate. (by the way, "registered independents" should not be confused with "Independent Party" which is a third party that tends to be far right aligned. The term for voters who don't side with any party is different in each state, but usually the terms "unaffiliated" or "independent" (no capital I) are used.). So usually when polls show that this unofficial party is opposed to something, politicians concerned with keeping their seats (or in Pelosi's case, keeping the seats of people in swing districts), will tend to consider this vote over the party line. Since the unaffiliated/independents and Moderate Democrats are more opposed to impeachment and are more critical to keeping the House in opposition control.
It's important to note that although Impeachment is an infrequent event in the United States, Impeaching the President is extremely rare, only two presidents have gone through the full process (Nixon famously resigned when it became clear that articles of Impeachment were being drawn up).
There are a lot of rules and procedures to the impeachment process, but there are three phases. Any office in government that is not a legislature position can be Impeached, not just the President and Vice President. In fact those are the only two officers who are elected officers that can be impeached. The other offices are typically Cabinet positions or any presidential appointments executive officers of the current president that are subject to advise and consent of the senate and any federal judge regardless of which judge appointed him (Federal Judges have lifetime appointments, and usually will either die in office or retire (especially if they side with the current party in power).). One is Impeached for "High Crimes and Misdemeanors" while in office. It's important to point out that Impeachment is not a criminal matter so if the crime has jail time, that's decided by the regular courts. Impeachment just removes the officer from his or her post. It's also important to note that while the offense ought to be a crime, the unwritten rule as to what is an impeachable offense is "Whatever Congress says is impeachable". If the President has unpaid speeding tickets or Congress thinks he's ugly, it's impeachable (they usually aren't this petty though).
We are currently a very likely phase one (more on this in a minute), which is an impeachment inquiry. This means that the House (and only the House as they alone can initiate impeachments) is currently looking for evidence that indicates the suspect did the crime they are accused of and if that crime is sufficient to rise to impeachment. What this means for the nation is very little. There is presently numerous comittees in the house looking at the President's actions for any oversight issues. Now they're looking at it with a recommendation to impeach. While I said Impeachment is not a criminal matter, if we want to line it up with a criminal case, this would be the police investigating someone for a crime. The inquiry will end with the drafting of Articles of Impeachment and the committee charged with the drafting voting to send the Articles to the Floor for a full vote. If they vote against, the whole matter shuts down and we go no further. If they vote in favor, then we go to phase two. At time of writing, Trump hasn't formerly entered this phase, as proceedings officially start with a referral to comitee. This hasn't yet happened, but Nacy Pelosi says she will be doing it, so this will change, but for now Trump is loosley in phase one. A total of 21 people in U.S. History have entered this phase formally.
Phase two is a pretty brief phase as rules of the House hold that any Articles of Impeachment on the floor will trump buisness of the day and must be scheduled in a very quick manner (I believe from introduction to the floor, the vote must happen, even if the Speaker wants to block it, and it must happen within three days of being recommended to the floor.). Each Article of Impeachment is voted on and those that pass the house are formal charges, while those that fail to pass are dropped. If any single Article passes (by simple majority 51% in favor or more), the officer is said to have been impeached. The House will then name it's Managers (Essentially, they are a team of Representatives who will represent the House's argument in Phase 3. If the committee is the police, these are the prosecutors doing doing) either by resolution naming the managers or resolution letting the speaker name them. 19 people have been Impeached in U.S. History (by the Federal Government).
Phase 3 begins with the Managers reading the successful articles of Impeachment before the Senate. The Senate will at this point debate to choose to accept some Articles but not others or none at all. To date, only one person impeached (Senator William Blout, the first person and only legislative officer to be impeached) has had all articles rejected (Since the Senate had expelled him the same day the Managers presented the Articles to him. Since Blout was not a civil officer, the debate was largely over the question of should he be impeached, with the conclusion being no removing all Congressmen/women from being impeachable officers). Once the articles are accepted, the senate holds a trial. If the Impeached officer is the President, then Constitutionally the Chief Justice of the Supreme Court will preside as the judge. Otherwise it will be the Vice President (in his role as President of the Senate, it's one of the few office powers he has, the other being "wait until the President dies") or the President pro Tempore of the Senate (aka the guy who keeps the Vice President's chair warm cause he doesn't have to be in the Senate Chamber unless he is casting a tie breaking vote). No one's really sure who presides if the Vice President is Impeached, with the nerds debating this between Chief Justice or President pro Tempore. The Constitution makes no recommendation and since no Vice President has even had a formal impeachment Inquiry because they don't do a whole lot of anything. The trial plays out like a standard U.S. Trial you see on TV with the noticable distinction of a much larger jury (all the Senators for a Presidential Impeachment will attempt to be present, if they have to wheel their death beds into the chamber.). Once the trial has reach closing remarks, there is a closed door debate and then the Senators Vote. It takes a 2/3rds Super-majority (67% vote in favor) of those present to convict on each Article of Impeachment. If the whole Senate is there, that's 67 votes for, but if Senator Death Bed croaks, the number adjusts to reflect. Of the 19 Impeached, 7 were aquitted of all charges, 8 were convicted and 3 resigned office or were otherwise removed before reaching the final vote, prompting a dismissal of the charges. A conviction on any charge removes the officer from office but does not necessarily bar them from holding office, though the Senate may vote on this additional punishment at a later moment (the primary purpose is the immediate removal so they're not going to have this debate on trial).
Of all individuals beginning the formal process (21) 15 were federal judges, 4 were presidents, one was a Cabinet Secretary, and one was a Senator. No executive officer has ever been removed via impeachment. One senator and two judges were had their charges dismissed. Of the four Presidents with formal inquiries, two were impeached (Andrew Johnson and Bill Clinton). Richard Nixon resigned upon learning Articles had been drafted by comittee but before they were voted on on the Floor of the House. Since removal from office stops the process, he was never impeached. James Buchanan had no Articles recommended to the floor, but the committee did say he was a very naughty boy (my words, not theirs. They said he was the most corrupt President in the country's history up to that point (1860) and the November election of President Lincoln was in part with the Country being disgusted with Buchanan and his party.). Johnson would be the closest the nation ever got to impeaching the President, surviving by just one vote.
Politically, Impeachment of the President is very risky politically as the action doesn't hurt the party in the long run and both Presidents who went through all three phases, both were vindicated (Modern Republicans are more likely to say Clinton's impeachment... if you get them behind closed doors and loosen their lips with a few drinks you'd probably hear this from some hardliners too. The law Johnson violated was eventually found to be unconstitutional by the Supreme Court upon proper challenge and Johnson had deliberately broke it because he wanted to get a court ruling.). In the case of Buchanon, it might have played a part in the fall of the "Jacksonian Democrats" (The first Democrats, and at the time a very powerful party. From 1828 to 1860, only four Presidents were Whigs, Two of whom died in office (William Henry Harrison, the most notable Whig President from this period is only notable because he was the first president to die and was in office for all of 30 days before he died!)). Buchanan isn't discussed much as much because the nation's lowest point in it's history was the five years following his Presidency and successor Lincoln is generally seen as one of the Greatest Presidents in U.S. History (with only George Washington being the legit challenger to that title... and most Americans will be content to say they tied rather than debate against one of them.). If Buchanon's legacy is brought up, it's often in a way that will frame Buchanan as the Neville Chamberlain to Lincoln's Winston Churchill (a weak willed leader who tried to avoid the inevitable war followed by the reluctant war leader who held his nation together in it's Darkest Hour. Nevil Chamberlin did at least get a memorable quote to his weak attempt at a peace at all costs, so he has that over Buchanan.).
Nixon is a rather odd place in American History, being a good candidate for the worst President of all time (yes, he ranks lower then the guy who died with a month in office) and at the same time, one of the best Presidents of the Cold War era. Even as the Watergate Scandal was just starting to crack, Nixon won every state bar Massachusetts and D.C. in his re-election bid (and even then, those two electorates were very narrow votes). And he's also famous for adding the suffix "-gate" to every scandal's topic since. The general thing that Americans seem to reflect upon is that Nixon was wrong for ordering the break-in at the Watergate Hotel, but he was impeachable for covering it up when accused. "The Cover Up is worse than the Crime" is a oft repeated addage in the United States.
This is probably the reason why Trump is releasing the materials request of him unusually fast in how these scandals play out. It presents to those who are defending him that, since he's not hiding anything (I know there are some who don't see the transcript as sufficiently opened, but there's a equally vocal amount of people who see it as valid. For purposes of informing, I will only say that the document can be used to support either arguement, and that in absences of more evidence, I am trying to remain neutral to the whether Trump is impeachable or not). In fact it is the fact that Trump is being open about a document that his critics are using him against him that actually lends to his favor.
In law, when evaluating evidence, if a testifying witness says something that reflects poorly on him as a person, it's actually considered more likely to be an honest statement then a simple denial (in fact, certain testimonies can only be admitted if it's not in the witnesses best interests to introduce it, namely, a statement which is not your own (hearsay) and you are quoting). If I am on trial for shooting two law officers, the statement of "I shot the sheriff, but I did not shoot the deputy." Is considered more honest then saying "I didn't shoot anyone" because you're admitting to a crime you did commit while denying a crime you didn't.
In the case of Trump, the accusation is basically "Trump threatened to withhold money unless the Ukrainian President re-opened the Biden Investigation." In the transcript, Trump does indeed ask the Ukrainian President to look into the matter (I shot the sheriff) but never mentions the money in the call, let alone in connection to the request (I didn't shoot the deputy!). Yes, it's not something that looks good, but it's not as bad as the crime he's accused of.
At this stage, the chess board is that both players are making their gambits. The Democrats (led by Pelosi) are sending a message that they are willing to risk losing the center, and possibly the 2020 election, in order to remove Trump because they truly believe Trump has committed a criminal act, while Trump is trying to capitalize on this shift by playing on the unpopularity of Impeachment and the Weakness of the case against him. If we want make a chess analogy, Pelosi has sacrificed a queen (the unaffiliated/independent) for what she hopes will be a quick checkmate, while Trump has made a seemingly bad counter-move (releasing documents with damaging information), but if he can survive, he could exploit Pelosi's opening move. Pelosi does have to "Mate in 14 months (the next presidetial election)" in order to win. Ultimately, with no available polling on the Unaffiliated/independent's general attitude towards these moves, it's too early to tell who will be more likely to take the king. But it's clear someone will be in check and possibly mate very soon.