Emulation isn't only for PC anymore. You can now play many emulators including MAME on nearly every game console. This usually requires the console to either have a modchip or to be soft-modded (software-hacked). Here is some basic information to on emulation on game consoles.

Original Xbox:
CoinOps is the ultimate emulator for the original Xbox console! It plays MAME arcade games and a whole host of console titles as well, all with a gorgeous frontend. It even has trackball support for the X-Arcade Tankstick, you will just need a USB-Xbox adapter cable. This will require your Xbox be soft-modded or have a modchip in it.
Downloads for CoinOps

CoinOps FAQ

X-Arcade Theme

The Nintendo GameCube requires a modchip to use emulators. If modded you will have to put your games on the very limited 1.5GB disks it can read, which makes it pretty limited. It can, however, run several console emulators and MAME.

Emulation on GameCube

To use emulators on the Wii you will either need a modchip, or you will need to do a softmod. The Nintendo Wii has many emulators which can run on a disk or from an external hard drive, giving you plenty of space for your games. Original Wii consoles also have GameCube support, so you can also use the GameCube emulators on the Wii with full-size DVD disks.

Emulation on Wii

Xbox 360:
If you hack your Xbox 360 console at all, you will not be able to take your console online at all anymore, you will get banned. Many people have had their consoles banned, and have turned them into emulation machines.

More info to come.

PlayStation 3:
The PlayStation 3 requires a mod to take place which will inhibit you from ever getting any firmware updates, which also means you won't be able to take it online.

Emulation on PlayStation 3

PlayStation 2:
The PlayStation 2 console will require a modchip or softmod to play emulators.

Emulation on PlayStation 2

The good thing about playing emulators on the Dreamcast is that you don't have to mod the console in any way. The downsides are that 1) the console has very limited power, so it is very limited on what games it can play correctly, 2) you have to burn the games to CD, so it's not easy to just add and remove games like you could with a hard drive based console and is very limited in space, 3) there is very limited support for emulation on Dreamcast these days with much better options on other consoles, and 4) you have to find the ROMs for the old version of MAME that DC emulators use. Considering those limitations, however, it certainly may make your dusty unused Dreamcast useful for something.

MAME: MAME4ALL seems to be the most capable emulator for MAME.

NES: NesterDC does a great job at playing NES games on your Dreamcast.

More Emulators for Dreamcast