The river is the name given to the final card dealt to a player, or to the community cards, in a hand of poker. Most often, and without any other context, it usually is defined as the final card dealt to the community cards in a game of Hold 'em.
Other forms of poker use the term river to describe their final card, as well. In 7-Stud, the "river" is the final card dealt to each player (so each player has their own "river"). In five-card stud, the "river" is the fifth (and final) card dealt to each player.
In a more general sense, the river refers to the final play of a hand. Players speak about "lasting until the river", or "making it all the way to the river", in which case they are referring about making it to the showdown and the last play of the hand, and aren't necessarily talking about the final card itself.
River is also used as verb (in the same sense that flop is used both as a noun and verb). Like "flop", it can be used to describe making one's hand with the final card where the hand being made is the object of the verb (e.g. "I rivered a flush!"), although it is most often used with a competing player as the object of the verb (e.g. "I rivered that guy!"), to describe the act of completing a draw with one's final card and beating a previously-winning hand. This may or may not be a bad beat in the classic sense, but the player with the previously-leading hand certainly won't be happy if someone rivers them.