I recommend wikipedia's "List of forms of Government" as a springboard. It's not going to get you to five, but there are ways to get you there with specific country examples. Wikipedia generally has four levels of organization of any government, and a myriad of sub forms.
This is probably the easiast as the terms are easy to understand and you can go from the three categories to five by splitting the most broadly defined ones into their two categories. This divides a government broadly under who has the power in a government body and the constraints put onto the body. The three forms are:
Democracy: (citizens have equal say in government decisions).
Autocracy: (A single individual... or a small group of a leader and advisory circle, have a say in government decisions).
Oligarchy: (A body of citizens with special knowledge has control over the government direction.).
The first too can be further broken down as follows:
Democracies can be either Liberal or Illiberal. A Liberal Democracy is characterized by restraints on the laws that can be passed that are designed to protect the minority from a political "Majority Rule". Illiberal Democracies do not have these protections in place, creating a Democracy where the political majority can oppress the political minority.
Autocracies can be further refined into Dictatorial Autocracies and Monarchial Autocracies. In both cases, one person makes the laws, but the key difference is who do we determine gets the job when the current person who has it dies. Monarchial Autocracies (often called Absolute Monarchies) give the job to a hereditary line of succession, usually the oldest male (though some systems are oldest child, regardless of success) and then go through the line through that country's own succession lines (usually a child of the next in line will be favored over a sibling of the next in line). Dictatorial Autocracies don't limit the new law maker to just the current one's family (though it can be, the kid is just as likely to get the job as the adult).
While Oligarchies have a wide range of subcategories, these are all based on what body is allowed to make the choice. Theocracies are common though Technocracies use scienctists and Meritocracies use "best qualified for the job".
For examples of countries with these lines:
Liberal Democracy: United States of America.
Illiberal Democracies: Turkey (If you vote against the current party, you go to jail), maybe Russia (prone to election fraud favoring the current party).
Oligarchy: Vatican City (The Pope is elected by the College of Cardinals, who are all the highest rank within the rank of Bishop in the Catholic Church, a Theocracy. A cool sub-national one is the town Disney World is in, called "Reedy Creek" or "Reedy Creek Improvement District" which allows only residents, and the only residents are employees of Disney World (and high level at that). This is a Corporate oligarchy and a Timocracy (land owners only). Most companies that have stockholders are also Corporate Oligarchies as you must own part of the company to vote... with the vote weighted to your share of stock.
Dictetorial Autocracy: North Korea is the best example, but that's debatable. Their view is that the leader retains leadership after death, thus the past two office holders didn't inherite the role but are representing him.
Monarchial Autocracy: Saudi Arabia, where the head of the royal family is the government. Do not use any European Country with a king or queen. They aren't absolute in power.
As above, this definition is nebulous and you have to understand the country that in general... Like I said, all European Monarchies are not Monarchial Autocracies... but you still can't choose who is the Queen of England. Usually the European Monarch is a Head of State while the Head of Government is an elected official. The United States citizen can get confused by this because the Head of State and Head of Government are the same guy (The President) and it's hard to catagorize the job of each, because each nation sets the role. These governments are typically described in one of four catagories:
Absolute Monarchy - Monarch is head of State and Government (Saudi Arabia)
Constitutional Monarchy - Monarch is head of State but not Government (United Kingdom, Japan.)
Semi-Presidential Republic - Head of state is not a Monarch and not head of Government (France, Russia).
Presidential Republic - Head of State is Elected and is also Head of Government (United States of America)
As a rule, a nation is a Republic if it does not have a Monarch as head of state (Again, going by North Korea's rules, It is in fact a Republic as you can't inherit leadership).
A better way to go is not power source, but power organization. This classifies governments by level of organization of territory and what powers each region has with respect to national government. This one has four, but there's a lot of weirdness in one to carve out five... and if you want to be creative, there can be six or seven.
A Unitary State- A nation where the national level government is the only body that can make laws for a subnational region. Local government is administered by appointed officials to make sure government services are performed. (France is a Unitary State). The Central Government is the sole government.
Devolution (aka Devolved Unitary State): A Unitary State that permits some form of subnational government to exists. What powers the devolved states have are dependent on the National Government and the National Government may lay claim to powers it previously let the sub-national government have. (The United Kingdom is a Devolution State, where Scotland and Northern Ireland are given their own regional Parliment by the National Parliment... which also the Parliment for the subnational regions of England and Wales). The Central Government creates the Sub-national Governments.
Federation - A Federation is the reverse of a Devolution. In this case a bunch of small nations enter into an agreement to have a collective Central Government with limited powers. What these powers all depends on the nation, but essentially, the Central Government may only make the laws with the powers it's member-states ceded... but the many states still retain law making abilities and the Central Government cannot take it away. Again, those powers are normally agreed upon by the members and any new members would cede those powers... but it's not universal. For example, two nearly identical Federations, The United States of Mexico and Switzerland ceded identical powers (printing money, inter-state authority, International authority) the U.S. States also ceded Immigration Authority while the Swiss Cantons (their word for States) did not. So a U.S. Citizen is determined by the National Government, while a Swiss Citizen is someone who is a citizen who can be declared a canton citizen. This means that the laws are not universal across the nation and you can get in trouble in one state for one law violation, but be okay in another state because it isn't a law violation. In Federations, the National Government is Formed by the sub-national Governments.
A Confederation (or confederacy) - Is a looser form of Federation... and the line can be difficult to assess... not to mention some Confederacies are Federations because back when they formed, "Confederacy" meant the same thing "Federation" does (The Swiss Confederacy is the official name for Switzerland, even though it's a Federation in classification purposes). For the technical description , a Confederacy is a government body where the Sub-national elements are equal in power with the Central Government. The Central Government may pass laws, but the sub-national governments must each pass the law in order for it to take affect in that sub-national region. Typically, they don't last long and are usually a temporary government arrangement while the sub-nations discuss a Federal arrangement. If it's intended as a permanent government, it tends to be short lived as well because the equal power between the National and Subnational Government naturally means the sub-national government can leave after joining (while not discussed often, during the Twilight of the U.S. Civil War, the Confederate States of America were nearly threatened by member states wanting to leave the CSA. The Union defeated them before this could have actually happened, but it was a problem that wasn't helping their side). Where they do succeed is if the powers are economical only, such as the only extant one under this definition, The European Union... soon to be down one UK, which exited legally, and even then the currency of the EU was not the adopted by all members as Bad Boy UK kept it's Pound Sterling while the others switched to the Euro. In Confedericies, the sub-national government is more powerful than the Central Government... and can leave when they want.
Finally, we have Common Wealth which are a group of nations that share a unified identity but no central government. Often they are nations with a common origin but for historical reasons, parted but are not upset over it and may still share national systems. The Commonwealth of England are a group of Nations that were all former colonies of the UK which were given full independence but still culturally identify with the UK. Some of the members go so far as to share the same head of state (The Monarch) but have different heads of Government (Queen Elizabeth is the queen of U.K., Canada, New Zealand, Australia, and several other nations in the Commonwealth) but its not limited as the Commonwealth of England does have Republic nations in it's membership. The United States is uniquely not a member, because they do not want the British Monarchy to be anything politically Important to the country where the Crown is the head of the Commonwealth Organization.
The sixth member of this group is an Empire, which is a country where the sub-nations exsisted before the National Government, but the National government has subsumed the role of the National Government. Typically this is done by the National Government being a victor over the sub-nation in a war of conquest against the sub-nation, though if they are both Monarchies, it could be that the Monarch is the rightful heir to both nations' Monarchies. The Hapsburg Empire is famous for using marriage to unite all sorts of European Kingdoms with the family having members on thrones throughout and at one point, even the Emperor of Mexico was from the Hapsburg House. Europe has a rather checkard history of monarchs who weren't from the nation they most famously ruled. Catherine the Great, was the famous Empress (regent, but her kid wasn't "The Great" so... ) of Russia despite being German and marrying into the Romanov family. Less great, Tsar Nikolai II's wife, Tsarita Alix, was also German, which was a bit akward for all of Russia when Nikolai had her run the day to day government of Russia while he managed the war... with Germany... The Russian people took some issues with the fact that they were fighting the Germans... while the German Monarchy was effectively in control of the home front... and thats why we have no more Tsars... well, there were other issues, but Nikolai and Alix weren't innocent when the revolution came...