This is a description of President (Pagat link) alongside the local rules that I played with a lot in college.
You need one french-suited (aka "standard") 52 cards deck (no jokers) and 2+ players. I consider 4 players the best, 2 to be bad, and I don't remember playing more than 6 (but the rules can support it).
The goal is to get rid of your hand as soon as possible. The first player to empty their hand ranks first, the second to empty their hand ranks second and so on.
Cards are ranked 3-4-5-6-7-8-9-10-J-Q-K-A-2, with 3 being the lowest and 2 the highest.
Each player is dealt one card at a time until the whole deck is dealt. Depending on the player count, some players may have more cards in hand.
A round starts with the current player playing any number of cards with matching values from their hand into the table, starting the "pile".
For example they can play a single 4, a pair of 4s, three 4s, or four 4s.
The next player (I think it's clockwise) may then play a group of matching cards of exactly the same size from their hand, with either equal or greater rank. They are not obligated to play and may pass even if they could have played something.
For example, on a single 4 you could play a 4, a 5, or a king; on a pair of 4s you could play the other pair of 4s or a pair of 5s, but you could not play a single 5 or three 5s.
This continues until one of three things happen:
When one of those conditions is met, the pile is discarded and whoever played the last card starts the next round. If that player emptied their hand during that round and left the game, then the next player in turn order starts the round.
2s are the highest ranked cards and immediately win you the round when played. You are still required to play the correct amount of them, for example on a pair of 5s you need to play a pair of 2s, you can't play a single 2 on it.
If you play a card with the same rank as the previous one (which only happens on single card rounds since on a pair it would end the round), then the next player gets a "ta gueule" ("shut up").
They are forced to either play a card of the exact same rank, pushing the "ta gueule" to the next player or ending the round if it's the fourth one, or pass. They may not even play a 2.
If a group of four cards is played, then it's a revolution. The rank order gets reversed (lowest ranks become highest and vice versa) and 3s get the power of 2s who loose it.
This lasts until the end of the round or another revolution is played (which can be on the same round but it's very rare that two players get one).
When a player empties their hand during a round, they immediately leave the game and get their rank. The first to leave the game ranks first, the second to leave ranks second and so forth. The game ends when all but one player empty their hands.
Once the ranking is established, the first player is given the title of "president", second player of "vice-president", second to last player "vice-asshole", and last player "asshole".
The "asshole" deals for the next game, but once hands are dealt and before the first round starts, those players must trade cards. I do not like this rule, at all, but that's what we played with.
The president gives 2 cards of their choice to the "asshole", who in return must give their 2 highest-ranked cards. The vice-president does the same with the vice-asshole but only 1 card.
The president then starts the first round.
This is a variant that I came up with to try to minimize the impact of ranking last (because I hate the trading rule since it just makes the same player keep winning).
It was a neat experiment, but it slows down the time between games a lot. In retrospect, just removing the card trading would probably do it better.
After cards are dealt, instead of the forced trade, each player puts two cards from their hand face down on the table.
Once all cards are placed, they are turned faced up, then each player in ranking order (first place goes first and so on) takes two cards of their choice from the pile to add to their hand.
Once all cards have been picked up, the game begins.