Basically Amazon Prime is a service that costs $79 a year. It upgrades your shipping on all orders to free 2 day shipping or pay $3.99 for 1 day shipping and you get a year free for being a student. You need a .edu email address.
To sign up for and use Amazon Student, you must (1) have an Amazon.com account, (2) be a college student actively enrolled in at least one course at a college geographically located in one of the 50 United States or the District of Columbia, (3) be able to provide proof of enrollment upon request and (4) have a valid e-mail address that contains the domain suffix .edu.
Edit: 12/26/2011
When your year expires you can renew it for $40 a year up to the next 4 years.