You have alot of options for governing your country in HOI 3. Each government has a political ideology -- maybe it's socialistic, or maybe socially conservative. Maybe it's a dictatorship, but not one that would get along with the fascists in Germany and Italy. Or maybe your leading a centrist coalition in a democracy, and you're hoping you'll make it through the next election without having your hat handed to you.
Hearts of Iron III does have elections in democracies. Other kinds of political instability can affect non-democracies. Popular support for the government is responsive to your actions as president or dictator or prime minister. The philosophy of your government determines who (which ministers) you can place in your cabinet, which is one way to get certain advantages for your country's economy or war production.
Politics inside your country, or someone else's, can be manipulated by friendly or enemy spies. You could send agents to undermine your enemy's government, or to lie to your people and prepare them for an aggressive war against an innocent neighbor.
There are a number of aspects of the game, including which laws you are allowed to enact, that depend upon what political party you have in charge, or on the amount of popular support you have. Don't underestimate the importance of keeping your population happy. If they give up on the war before you do, you may not have much to say about it!
Lastly, when you take over another country, you have to decide whether to be nice to them, and try to get their people to enlist in your army and join your "war of liberation" or to be ruthless in your occupation, squeezing the most you can out of the resources of the country to benefit your war effort.