The United States chooses its president and vice president through a process called the Electoral College. The Electoral College comes from from the constitution and it is a compromise between a popular vote by citizens and a vote in Congress.
Here’s how the Electoral College works: Citizens vote by state, and each state has a certain number of votes to give to a candidate of their choosing. The citizens’ votes often tell the state how to cast the states votes but the states don’t have to listen to the citizens at all. The number of votes a state has is determined by the states population, plus two. The plus two actually makes it so that the smaller states have a lot more power then the bigger states even though the bigger states hold a much larger percentage of the country’s population. Overall the Electoral College is very flawed.
In Russia, the president is elected on basis of equal, universal and direct suffrage through secret ballots. (I just got this part from Wikipedia because i don’t know a lot about how it works in Russia)
Sorry if my explanation isn’t any good, I hope it helps.