XBox Live is an online service available to original XBox and XBox 360 consoles. You must have an XBox to connect to this service. And, your XBox had better not be modified to play pirated games, or your console and your account will be banned for life.
There are two tiers to the online service: Gold and Silver. The silver tier is free and allows you to view the free content available, which is usually movie and game trailers, access to the marketplace where you can upgrade your membership, buy Microsoft points (online currency) to buy games, and download and play free demos of 
games. The gold membership is a paid membership which is about $50/year. You automatically get 1 free month with the purchase of your XBox console. Many games come with a 1 month free code. With the paid, gold membership, you have access to your Netflix account (if you have one) and can watch live, streaming movies from your Netflix queue, right there on your living room TV (no PC required for viewing). You also get access to your facebook account, twitter account, and the primary purpose of XBox Live Gold membership: You can play games with or against online 
gamers, “friend” online gamers, be able to brag about your online achievements, and be placed in global high score lists. There are more services than this and the list of services keeps growing. If you’re a software developer, you can even write your own games using XNA technology and publish your games on the XBox Live “Indie Games” section for anyone else to purchase. You get real money in your account.