Mappers, or Memory Mappers are hardware in cartridges that allow games to be bigger than the default size allowed by the system. This hardware is in addition to the other RAM and ROM chips in the cartrige. Ussually just microchips on a circuit board, but the Famicom Disk System and the Starpath Supercharger (for the Atari VCS) are also types of mappers.
The Atari VCS only supported 2k cartridges. Mappers, then consisting only of Bank Switching, were invented to load games up to 256k in size, 2k at a time.
The NES supported a 32K of PRG ROM, 8K of CHR ROM (or RAM), 2K of WRAM, and 2K of VRAM. NES hardware mappers easily allow these limits to be avoided by mapping or bank switching extra PRG ROM, CHR ROM, RAM, WRAM, VRAM, audio, and video data. In addition to larger games, mapping data to and from battery backed RAM is done with hardware mappers. There are many 256k NES games, a few 512k games, and at least one 1024k game.