Rummy

A.K.A. Western Mahjong.

Take a hand of cards, arrange them in such a way that they form "melds" (runs of 3 cards in a single suit, or 3 or 4 of a kind).

Formerly a very large family of games, Rummy has been whittled down to two games: Gin Rummy, and Canasta. Gin emphasizes getting all your cards into melds and has a fixed size hand; scoring centers around "deadwood", cards that don't form a meld, which score for your opponent. Canasta emphasizes making as many melds as humanly possible and has hand sizes that quickly get out of control. Gin only requires one deck of cards; Canasta involves four decks and a lot more rules, many of them centering around the red and black 2s.

Gin, because it's simpler and involves only one deck and two players (Canasta is usually played by four), is probably more popular.

Card Games
Collectible
Card Games:
Call of Cthulhu - Cardfight!! Vanguard - Fire Emblem Cipher
Force of Will - Jyhad - Magi-Nation Duel - Magic: The Gathering
Netrunner - Pokémon - Star Wars: Destiny CCG - Yu-Gi-Oh
Other
Card Games:
1000 Blank White Cards - 7th Sea - Apples to Apples - Bang!
Cards Against Humanity - Coup - Decktet - Dominion - Dvorak
F.A.T.A.L. - Mafia - Mag Blast - Mao - Munchkin
Race for the Galaxy - Sentinels of the Multiverse - Tanto Cuore
Traditional
Card Games:
Bridge - Cribbage - Mahjong - Patience - Poker - Rummy - Tarot