Xbox 360 is backwards compatable with certain games that were released for the original Xbox. Currently there are more than 200 BC titles with more being added all the time. Big name titles such as Halo, Halo 2, Splinter Cell: Chaos Theory, Star Wars: Knights of the Old Republic, and Ninja Gaiden Black are just some of the Xbox games you can play on Xbox 360. For the full list, check here
There is a requirement for backwards compatability and that is you have to have a hard drive for your Xbox 360. If you bought a Core system you can't play original Xbox games on your system until you pay another $100 for a hard drive. Which is why I suggest not getting a Core and sticking with the Premium system from the start if you can.
The way the backwards compatability software is updated on your system is done in a couple of different ways. If you have at least an Xbox Live Silver account, the update will download automatically when you try to play a BC game. The other way is you can go to the official Xbox website and either order a CD from Microsoft or download the update and burn it to a CD yourself. You then pop the CD into your Xbox 360 and it will be updated.
Besides it being really convenient to be able to play both Xbox and Xbox 360 games on one system, playing Xbox games on your 360 has other benefits. Each and every compatable Xbox game will be upped to 720p/1080i resolution (assuming you have an HDTV) and will take advantage of full screen anti-aliasing. This makes the games look really, really good. The improvement is actually good enough it almost feels like a whole new experience so it can definitely add new life to your older Xbox games.