The short version of this answer is that it appears that Trump may have been trying to get a foreign nation to collude with him to help him win the upcoming election via undermining his political rival, and he by abusing his power as president to try to make decisions that should be made in the best interest of the government based off of rather Ukraine would help in his personal desires. His white house then allegedly worked to undermine any attempt to investigate the potential collusion.
This is confounded by the fact that Trump has been investigated for much of his presidency for...colluding with a foreign government (Russia) to aid him in winning an election; and for abusing his power as president to prevent anyone investigating him for the act.
The fact that this action draws such a parallel to the larger Russia collusion allegation makes the claim worse, as it arguably provides further credence to the allegation that Trump previously colluded and conspired to hide the fact if it can be shown he is doing much the same thing currently.
However, the biggest issue is the proof. A transcript was released from the whitehouse with the full call. Since the white house released it we can generally take the transcript to be honest, or at the very least it's unlikely that the transcript was doctored to make things look worse for Trump*. As such the issue of proving the claims is pretty cut and dry, he definitely did say the things in the transcript; one can not claim false news or biased investigation to refute those statements. Allegedly the conversation itself is damning enough to prove intent to conspire with a foreign government by itself and thus is a 'smoking gun' for the charge of attempting to blackmail a foreign government into helping to make his political rival look bad. Proving, or disproving, the claim that the whitehouse actively tried to hide the evidence of the call is not as easy to do, but the is at least some evidence that may back up such a claim.
- as an aside some have implied that they may have doctored it to not be as damning, based off of evidence that the white house has doctored comparatively more minor details to make Trump look better in the past; but there is little evidence of this claim and I personally consider it unlikely the transcript was altered; in either case we can at least presume that it was not altered to make him sound worse.
To better demonstrate the claims and controversy I'll quote the transcript directly. There are many potential sources for the original transcript of the call, I'm using this one simply because it was the first I found that allowed easy copy and pasting. I'm needing to clean up some copy and paste errors converting from PDF, so please forgive me if I miss any odd formatting or typo's in these quotes.
Trump attempting to get Ukraine to look into the Russia hacking:
This is arguably the least damning of the allegations, but it happens first chronologically on the call, as such I'll address it first here. Trump asked the Ukraine president to look into the Cloudstrike, the firm that investigated the Russia hack into the DNC.
Trump first starts with a rather long discussion of how much US does for Ukraine and how US does more then most other countries, I won't quote it due to it's length and it's pretty clear what is being said. The next time he speaks he starts out with this:
I would like you to do us a favor though because our country has been through a lot and Ukraine knows alot about it. I would like you to find out what happened with this whole situation with Ukraine, they say Crowdstrike ... I guess you have one of your wealthy people... The server, they say Ukraine has it.
Shortly later he says he would like the Ukraine president, or perhaps a delegate, to talk to the Attorney General about this:
I would like to have the Attorney General call you or your people and I would like you to get to the bottom of it.
Trump never gets into more detail as to what the favor is or what they should get into the bottom of. The original whistleblower's claims, seen here, makes a claim presumably alluding to this part of the conversation, that this constituted the president asking Ukraine to:
assist in purportedly uncovering that allegations of Russian interference in the 2016 U.S. presidential election originated in Ukraine, with a specific request that the Ukrainian leader locate and turn over servers used by the Democratic National Committee (DNC) and examined by the U.S. cyber security firm Crowdstrike, 3 which initially reported that Russian hackers had penetrated the DNC's networks in 2016
This requires some inference, since Trump never specifies what he is hoping Ukraine will discover by looking into things, thus making it the weakest claim against Trump. However, it should be noted that this came right after Trump stressed how much the US did financially for Ukraine. It's alleged this constituted Trump implying he would withhold this aid if Ukraine didn't help with this or his later request.
Assuming we used the original whistleblowers interpretation of this conversation this would constitute Trump trying to undermine the Russia investigation into hacking, which Trump was accused of colluding with Russia with. This would thus potentially be seen as his trying to remove Russia from the spotlight and thus remove any reason to believe he colluded with Russia. One could even argue this constitutes obstruction of justice, since it's unlikely Ukraine could provide factual evidence that Russia was not involved in the hacking; which would obviously be a problem since Trump already is facing some potential trouble due to other obstruction of justice allegations. Though this all requires reading quite a bit into Trumps words so I wouldn't call this allegation a 'smoking gun'. Moving on...
Trump wanted Ukraine to speak to his personal Lawyer
Trump also requested that Ukraine speak to his personal lawyer. This would be damaging since there is no reason his personal lawyer should be involved in affairs of the state. The only reason for someone who represents Trump the person, not Trump the president, to speak with Ukraine is if the favor that Trump was requesting was something that would benefit Trump personally, not the nation he is representing as president. Any such request for favors from a foreign goverment to do something that would benefit Trump personally can be seen as abusing his power as president when he should only be acting in a manner that will benefit the country as a whole; especially if one interprets Trump's favor as being contingent on the US providing financial aid as he would then be 'buying' this favor with goverment money.
However, to be fair it was actually Ukraine president who first brought up Gulliani. Moving on chronologically the President of Ukraine responded to Trump's request for some uncertain favor with cyberstrike with a rather general message of "of course our nations should aid each other", followed by seguing into talking point about his assigning a new ambassador to the USA. As part of this conversation the president of Ukraine, He then mentioned this:
I will personally tell you that one of my assistants spoke with Mr. Giuliani just recently and we are hoping very much that Mr. G1uliani will be able to travel to Ukraine and we will meet once he comes to Ukraine.
Trump thus is responding to the President of Ukraine's mentioning meeting Giuliani, saying:
Mr. Giuliani is a highly respected man. He was the mayor of New York City, a great mayor, and I would like him to call you. I will ask him to call you along with the Attorney General. Rudy very much knows what's happening and he is a very capable guy. If you could speak to him that would be great.
If this were the only statement it could easily be seen as simple pleasantries and nothing more. However, this is only 1 of 3 requests for Giuliani and the Attorney General to speak. Immediately after Trump requests investigating Biden and his son (see below) he says this:
I will have Mr. Giuliani give you a call and I am also going to have Attorney General Barr call and we will get to the bottom of it.
This time the request to speak to Giuliani is directly tied to a desire to have Giuliani involved in looking into a political opponent. This one does not sound like simple pleasantry, but a honest desire for someone who represents Trump, the person's, best interest involved with a request for something that would directly benefit Trump looks far more damning.
Trump reiterates one more time the desire for Giuliani and the Attorney General to speak right before the closing of the phone call.
The request to look into Biden
Now we are getting into the more damning aspect of the conversation, which happens very soon after the first mentioning Giuliani. Trump asks that Ukraine look into Biden and his son. To give context Biden had recently bragged that he got a prosecutor removed from office by threatening to withhold federal aid if it didn't happen. That prosecutor had been assigned to investigate corruption cases, and one of those involved a company Biden's son was associated with.
There have been claims that Biden had the prosecutor removed from office to protect his son. However, There are counter claims that the US ambassador had already desired the prosecutor removed because he was not doing enough to prosecute corruption and that Biden was thus acting in government interest to push for more active prosecution of corruption; it's even been argued this could put his son at greater risk of being investigated. Since Biden's guilt or innocence is not directly relevant to this topic I won't get into it in detail, but here is a skeptics question covering the key facts:
Trump's exact request was:
The other thing, There's a lot of talk about Biden's son, that Biden stopped the prosecution and a lot of people want to find out about that so whatever you can do with the Attorney General would be great. Biden went around bragging that he stopped the prosecution so if you can look into it ... It sounds horrible to me.
This is far less ambiguous request then the one about Crowdstrike. It's hard to read this as anything other then Trump asking Ukraine to look into a criminal investigation of Biden's son, and considering that Trump said he thought it sounds horrible it seems rather clear Trump believes, or wants, that investigation to yield damming effects. The fact that he wanted his personal lawyer involved with this further suggests he believed this investigation would benefit Trump personally.
Regardless of rather Biden is or is not guilty of anything this is requesting a foreign nation's help in what Trump clearly believes will make Biden look bad. Considering Biden is a clear political rival and elections are about to come up it's believed by many the primary reason for this request was to get enough dirt to make Biden look unattractive to voters to help ensure he can not beat Trump in the election.
Furthermore seeing as this all came after the discussion of all the aid the US provides to Ukraine one could interpret this as implying that he may withhold financial aid if Ukraine does not help him could.
Any attempt to request another nation act in a way that would hinder a political rival would be undesirable in any situation. However, coming on the heals of the allegations and long investigation into rather or not Trump colluded with another foreign power (Russia) to interfere with a different political rival (Hillary) it is particularly problematic. It can be seen as showing a pattern of behavior, or even as further evidence against Trump's willingness to collude with Russia by demonstrating a desire to collude with another nation.
Final implication of potential tying help to money
Right after suggesting Giuliani be involved with the investigation of Biden and his son Trump adds this:
I'm sure you will figure it out. I heard the prosecutor was treated very badly and he was a very fair prosecutor so good luck with everything. Your economy is going to get better and better I predict.
This has also been interpreted as a backward implication that financial aid will be provided, to help the Ukraine economy do better, if the country helped with the Biden investigation; though it is rather ambiguous and I'm sure Trump supporters and Trump haters will read different meanings in the statement.
Withholding of evidence
The original whistle blower made the claim that the White house considered this call damaging, saying:
The White House officials who told me this information were deeply disturbed by what had transpired in the phone call. They told me that there was already a "discussion ongoing" with White House lawyers about how to treat the call because of the likelihood, in the officials' retelling, that they had witnessed the President abuse his office for personal gain.
He then goes on to describe the white house allegedly hiding the call in an overly classified system to prevent others from witnessing it.
II. Efforts to restrict access to records related to the call
In the days following the phone call, I learned from multiple U.S. officials that senior White House officials had intervened to "lock down" all records of the phone call, especially the official word-for-word transcript of the call that was produced—as is customary—by the White House Situation Room. This set of actions underscored to me that White House officials understood the gravity of what had transpired in the call.
• White House officials told me that they were "directed" by White-House lawyers to remove the electronic transcript from the computer system in which such transcripts are typically stored for coordination, finalization, and distribution to Cabinet-level officials.
• Instead, the transcript was loaded into a separate electronic system that is otherwise used to store and handle classified information of an especially sensitive nature. One White House official described this act as an abuse of this electronic system because the call did not contain anything remotely sensitive from a national security perspective.
Of course these claims are dependent on rather you trust an anonymous whistle-blower. If the claims are true this would be an illegal use of goverment systems, ironically similar to the misuse of government systems Trump attacked Hilary for. More importantly it can be seen as a preemptive attempt to obstruct justice, by preventing word of the call to reach those who would be legally required to investigate it.
By itself this call would be worrying to any president. However, it's similarity to the previous Russia allegations, with similar allegations, desire to have foreign nations undermine opponents, willingness to cover up the attempt later etc, make it all the more damning as it can be seen as a pattern of behavior suggesting a willingness to continue to tempt and accept foreign nations aid; which can be of great concern with the government predicting Russia will make attempts to interfere with the upcoming election.
Trump also has a number of lesser controversies, such as The New York Time's allegations of questionable tax returns to shift money from Trump's father to Trump, and the allegations of misappropriation of campaign finances in the Stormy Daniels affair. With these allegations, and the arguably unresolved status of the Russia investigation (with the final result refusing to declare Trump guilty or innocent and stressing it was up to congress to decide how to proceed) Trump was already in a politically vulnerable position with some calling for his impeachment already. This particular call can thus be seen as a potential tipping point for those already on the fence about impeachment.