When the Game Boy was released in the U.S. in August of 1989, it had three big advantages. First, it came bundled with Tetris, the Killer App. Second, with the console at $99.99 and cartridges at $19.99 the price was right. Third, the battery life of 20-35 hours was very impressive, especially compared to the competition (Sega Game Gear, Atari Lynx and NEC Turbo Express), which could barely muster 6 hours. The only disadvantage to the Game Boy was its four-shade monochrome screen, but compared to its predecessors, the Microvision, Game & Watch and Tiger Electronics toys, it was a huge leap forward. That drawback was not enough to keep people from buying millions of Nintendo's handheld system and few of the competition's handheld systems.
1. Super Game Boy
Nintendo planned to introduce a successor color console as soon as was practicable, but the monochrome Game Boy could not be quickly replaced. It took some years before LCD screen technology produced an affordable and practical color screen. Nintendo's first attempt at color was an add-on for the Super Nintendo, the Super Game Boy. The SGB could play GB games on a SNES, with SGB providing GB hardware and the SNES providing controller input and audio and video output. Any GB game could have a 4-color palette set from 32 choices or the user could make up his own palette. The user could also A set of 9 borders created by the SNES's graphics chip were available. The borders could have up to 64 colors. Nintendo's non-SGB enhanced games like Tetris and Metroid II would pick a default palette from these 32 choices. See here : http://tcrf.net/Super_Game_Boy
However, GB games could take advantage of some of the SNES's features, such as the SNES's sound chip, input from the 2nd controller port, custom borders and an ability to provide more color to the game than the built-in palettes could provide. While the GB itself had three palettes of four shades available to the background and sprites, the SGB could not colorize these directly. Instead, it could apply a 4 color palette to every 8x8 pixel tile. 4 color palettes were available with one common color. Thus at any time, up to twelve separate colors could appear inside the Super Game Boy gameplay window.
Beginning with Donkey Kong, games used the Super Game Boy to apply more colors than the built-in palettes. However, the most colorful screens were typically static screens. Apparently the SGB method colorization had problems tracking sprites, so screens with sprite movement would typically use only one or two of the palettes. Most games released after Donkey Kong would provide some level of Super Game Boy support, but eventually the support was limited to palettes and borders. These borders and custom colors would not be seen on a Game Boy Color, Advance or Player.
There are games that saw a re-release for the Super Game Boy with support for SGB features. Centipede, originally released separately, was later released in a multicart with Millipede, is one such game. Super Breakout was similarly updated when released in a multicart with Battlezone. Asteroids and Missle Command were released separately as pure GB carts, then in a multicart with SGB support. Tetris 2 was released first as a GB game, but quickly updated with an SGB compatible version.
2. Game Boy Color
The Game Boy Color supported a color screen and had a 15-bit palette of colors available. Each background 8x8 tile can select from 8 palettes with 4 colors each, and each 8x8 sprite can select from 8 palettes with 3 colors each. Thus up to 56 colors can be shown on the screen at any one time. However, this functionality is strictly limited to GBC games.
When the Game Boy Color was released, Nintendo allowed for some ability to colorize GB games. When a GB booted up, the GBC would apply either the default palette entry or a palette entry customized for a particular game or set of games. The boot ROM of the GBC would compare the hash of the ROM with a table and if there was a match, it would apply a custom palette. While there were over 1,500 GB games, including variations, the boot ROM table only had entries for 84 of games. Nintendo's own titles always got an entry, but some third party games also received some entries. Weirdly, there are several games that will use a custom palette only for their European version. For example, Mega Man 1,2 & 3 will show a custom palette with Mega Man in blue, but only if the European cartridges are inside the Game Boy Color, Advance or Player.
If there was no match, then the GBC could be told to apply one of twelve palette entries by pressing the A or B button, with or without a directional on the Game Boy Color's boot screen. The player would have to be quick to do this, otherwise the game would load with the default palette.
This colorization scheme allowed for separate 4-color palettes for the background and two 4-color sprites palettes. Thus up to 12 colors could be available, but sprites tend to use transparency for one color, thus typically 10 colors are used. Because the sprites and backgrounds can be directly colored, the GBC can typically show more color than the SGB. Sometimes, the custom palette would have a detrimental effect. In Super Mario Land, World 1-3, there are tiles that will fall and hurt Mario. Because these tiles use a sprite palette, they are colored differently than the background palette, making them easy to spot. The color combinations and list of games that are supported can be found here : http://tcrf.net/CGB_Bootstrap_ROM
In order to provide software for its new system, Nintendo released several of its games with new colorization using the GBC hardware. Often these titles had the DX suffix after their names. Thus there were The Legend of Zelda: Link's Awakening DX, R-Type DX, Tetris DX. These games provided substantial enhancements over their older versions.
Other re-released games include Wario Land II, Joust & Defender, Centipede, Frogger, Legend of the River King GB, Harvest Moon GB, Prince of Persia, Space Invaders, (Classic) Bubble Bobble, Daffy Duck - Fowl Play, Looney Tunes, Ms. Pac-Man, Pac-Man, The Rugrats Movie, Speedy Gonzales, Super Breakout!, Titus the Fox. Pokemon Yellow, while it is a Game Boy Color game, hardware-wise, does little in terms of color beyond its SGB features.
Japan received a colorized version of Balloon Fight (Balloon Kid overseas) for the Nintendo Power rewritable Game Boy Pak. The game was never released as a standalone cartridge. Konami colorized most of its Game Boy games and released them in Europe in four Konami GB Collections. Volume 1 contained Castlevania: The Adventure, Gradius, Konami Racing (F-1 Spirit) and Probotector (Contra : The Alien Wars). Volume 2 included Block Game (Quarth), Frogger, Parodius and Track and Field. In Volume 3 you could play Bikers (Motocross Maniacs), Guttang Gottung (no idea what this is, it might be unique), Mystical Ninja (Ganbare Goemon), Pop'n Twinbee. Perhaps the best collection was Volume 4, consisting of Antarctic Adventure, Castlevania II: Belmont's Revenge, Gradius II: Return of the Hero & Yie Ar Kung-Fu. Castlevania II on this cart is interesting because the U.S./European GB version had the holy water as a weapon, but the colorized version found on this cartridge had the Cross, which was in the Japanese GB version. Interestingly, Capcom had intended to colorize its Game Boy Mega Man games and was going to release them as a Game Boy Advance cart, but the cart never materialized.
Finally, there are hybrid games that first saw a release with GBC support in mind. By allowing cartridges to be backwards compatible with the Game Boy while being able use the full color features of the GBC, Nintendo allowed its monochrome, 1989-vintage machine a few more years of life. The hybrid games are really GBC games with a black & white option. Thus the contrast may not be ideal where the color version uses dark backgrounds, as in Mega Man Xtreme. There may be odd color choices, like the white beard for Arthur in Ghosts 'N Goblins that makes him look like Santa Claus. The game may not run as smoothly because the GB runs at 4MHz and the GBC runs at 8MHz. Blaster Master: Enemy Below is one game that is sluggish on the GB where it is fairly smooth on the GBC (the NES game is smoothest of all). Graphical tricks may not be present in the GB because the GBC has the ability to perform "raster" tricks in hardware. Dragon Warrior Monsters 2 was the last hybrid game released, on September 15, 2001, thus giving the Game Boy (from July 31, 1989) twelve years and fourty-five days of continuous support. That is second only to the Playstation 2 and (perhaps) the Atari 2600.
The Game Boy Advance acts exactly like a Game Boy Color. The Game Boy Player for the GameCube acts like a Game Boy Advance, even though it is similar to a Super Game Boy. It does not support any Super Game Boy functions, but it does allow you to play Game Boy Color games on a TV.