The Atari 7800 is a third generation console by Atari, succeeding their ill-fated Atari 5200 however problems within the company meant that the NES overtook its launch making it a second place console in the US. It is famous as the very first console with backwards compatibility, being able to run Atari 2600 cartridges without the need for peripherals.
Rather like most models of the NES, the Atari 7800 has no RGB palette and because of this each emulator displays a completely different palette. Due to various conditions including which TV was displayed, what capacitors were used, how long you played the console for and even which variation of the Atari 7800 since there are more than one model of the console, there is no correct palette for the console and mostly it is personal preference. The closest to a correct palette what is called the Base palette that uses values directly from the console however the console has a warm up time and this affects a couple of games such as Choplifter where the palette changes the longer you play the game on real hardware.
To make matters worse, the PAL version of the Atari 7800 has a different palette to the US version and this affects all the games too, well ones that got a PAL version anyway. When ripping games from the console, it is advisable to use the NTSC palette as the default with the PAL palette also included as optional however there are a couple of exceptions. Plutos, Sirius and Pit-Fighter (all unreleased) were made in the UK and Jinks was made in Germany so the PAL palette would be the default instead.
AtariAge has over 200 palette options to choose from: http://atariage.com/forums/topic/210082-colorswhat-do-you-want/
Atari Rom Reader
This site already have the graphics of all 7800 games dumped, so ripping is as easy as picking a game and choosing the color palette however only one video mode is supported so as a result, these don't have all the graphics displayed. These include Summer Games and Winter Games where the main sprites aren't displayed however the backgrounds are, Pole Position II where the course select text isn't displayed. Rampart and Ms. Pac-Man don't display any graphics at all as they use a different video mode.
There is also a bug where the last column isn't displayed so you still have to check the sprites via an emulator just in case it is a part where it gets cut off.
Unlike many systems, Atari 7800 emulators do not have sprite ripping features and to make matters worse they aren't easy to capture either. It is because for some games the graphics are stretched out to fill a screen, a closest example would be like some Genesis games when played on Kega Fusion so they might have to be resized manually. They are lucky to even run some of the library to be honest.
ProSystem is an older emulator for the console however it runs most of the library. It can also select which palette you want. It seems to have issues selecting tracks in Pole Position II as well as the dipswitches don't work properly. Apparently the PAL version of Commando can not get past the title screen, whether it is a bug in emulation or the game itself is unknown.
It is also available as a RetroArch core.
MAME runs Atari 7800 games however there is apparently a timing issue with the games.
BizHawk used to use the EMU7800 core however the developers decided to do their own emulation for the console. It is a bit glitchy, has control issues and Tower Toppler doesn't work properly since the game instantly completes a level. Might improve in the future though.