There are a great many card games where the object is to get rid of your cards. These games are called "shedding" card games.
Shedding card games come in various shapes and sizes, and originate from a diverse range of cultures.
In Patience games, you rid your cards by placing them on stacks in immediately ascending or descending numerical order.
You probably know the ubiquitous solitaire game Klondike that comes with every Windows box. There are four stacks, one in each suit, going up from ace to king.
Spit is a patience game where there are no turns, and the object is to move your cards as physically quickly as possible. There is one stack for each player, and cards can go either up or down; suit doesn't matter.
In Cheat (also known by a more common name which is rude), you place your cards face down and announce how many you placed, but may lie about them. One stack, always ascending, but multiple cards may be played on each turn.
Additional games in this group include the proprietary games Flinch and Skip-Bo, descendants of a game called Spite and Malice. In these games, if you can't play, you place a card in a discard pile that will come back to haunt you.
Uno and similar games, such as Crazy Eights and Taki also use this mechanic, but introduce special cards that break the rules, allowing you to play multiple cards or out of order. These games also have a single stack, where cards either ascend or descend, or change suit.
Ladder / Climbing
Ladder or Climbing games also require players to play cards in numerically higher order, and not necessarily immediately numerically higher. However, in addition to single cards, pairs, sets of pairs, and other combinations can sometimes be played. When no one can play further, often the last player to play collects the played cards and starts again.
In Karma (also more often known by a rude name), each player gets 3 face down cards, 3 face up cards, and has to play cards onto the central pile in ascending order, although not necessarily the immediately numerically higher card. Various cards have special abilities that discard the stack or require a lower instead of a higher card.
Tichu is a Chinese variant climbing game, very popular among the gaming cognoscenti right now, mostly because it is a partnership game. Like most ladder games, the first card or cards played can be a single card, pair, multiple pairs, a full house, or a straight, and subsequent plays have to be of the same type, but higher.
Another game in this genre is Tien Len, which counts as its descendants proprietary games such as The Great Dalmuti and Frank's Zoo, as well as President (yet another game with an alternative rude name).
Melding games require you to be the first one out by combining sets of cards from your hands, either cards of the same number or straights of three of more numbers. The first one out may be the winner, or may simply get a bonus while the melds played count to your points.
Nearly all Rummy games, such as Rummikub, Gin Rummy, 500 Rummy, Tripoley, and so on, are melding games. In these games, you pick a card, try to form melds, and then (usually) discard.
Simple melding games include Old Maid, where you need only pairs, and Go Fish, where you need only fours of a kind.
Of course, there are always a few oddball games.
Mao is a game where the rules are kept secret during the game. Each player tries to play a card, and the person who knows the secret rule can penalize the player if they play (or do) something wrong. It's very silly.
Durak is a Russian game where you play cards by attacking the player on your left. Other people can add to the attacking cards, and you can either defend yourself by playing higher cards or pass the attack on to the next player by adding attacking cards. It's a bit of a free-for-all; cheating is explicitly permitted in the game.
Information about all of the games can be found on Pagat, Wikipedia, or Boardgamegeek.