The Game Boy is an 8-bit, 4th generation handheld console released by Nintendo in 1989. It had a monochrome display. Its successor, the Game Boy Color (GBC), was released in 1998. Unlike the original, it features a color screen.
Gambatte is hands-down the most accurate Game Boy Color emulator. It has a good range of options, though not quite as much as some other emulators.
BGB also has excellent Game Boy and Game Boy Color accuracy, as well as near flawless link support. It has a wealth of options for color palettes and even enabling Super Gameboy colors and borders, though it doesn't emulate all of its functions.
Visual Boy Advance doesn't emulate the color palette of the original hardware. Instead, it uses the color palette of a GBA, making Game Boy Color games look too saturated.
GameYob (on the DS) is the best GB/GBC emulator for that platform.
Higan is the only emulator that properly emulates all of the Super Game Boy's features, though the quality of its Game Boy emulation is inferior to that of other emulators, unless one hunts down an older version of bsnes, which used the Gambatte core for its Super Game Boy functionality.
GeMP and RIN have some incompatibilities with later PSP models. GeMP is incompatible with GameCategories folders.