To gain access to a program without authorization; To alter a program from its original or intended structure.
In game terms it means to "Enhance" a game either by obtaining things otherwise not obtainable in regular play, or by allowing abilities not normally allowed in the game.
On Animal Crossing Universal Codes were the popular form of hacking the game, mainly because anyone could do it, all it required was talking to Tom Nook.
Animal Crossing Gamecube
Animal Crossing: Wild World
Map Editor: http://aibohack.com/nds/ac_mapedit.htm