The online feature is Xbox Live, and it is probably the biggest reason I like Xbox. The number of Xbox Live players is MASSIVE and the online multiplayer has got to be the most fun I've ever had on video games. It also allows you to download games like Castle Crashers and Alien Hominid and such, along with downloadable content like Broken Steel for Fallout 3.
In regards to downloading things off your computer:
I haven't seen much of this, however I know that with Halo 3, you can download maps, videos, and gametypes off of Bungie.net.
Microsoft points are a sort of "currency" that you can use to buy items off of Xbox Live. You purchase it with real world money off of the internet or Xbox Live and they can then be used to buy things from the Xbox LIve Marketplace.