Amazon Prime is a service Amazon offers--for $80/year, you get quick free shipping from Amazon (Amazon only, and not other sellers at Amazon). You also get to watch a selection of TV shows and movies for free.
First, the shipping. $80 is a bit much to pay for free shipping if you're just a casual shopper, especially when other sellers often offer cheaper prices than Amazon. We regularly buy diapers and wipes from Amazon (they have good sales, and you can "Subscribe," get an even better deal, and then cancel the subscription once the item ships). I very occasionally buy other items that are on sale (or that I need on emergency basis). Not enough to warrant paying $80. But, students get Amazon Prime free for a year (minus free TV/movies). And after my free one year subscription ran out, they offered me another year for half price. $40 for free shipping and free TV/movies. I did the math, and it's still not a bargain, although I think we'll come out a little bit ahead with all of the diapers we buy. What about the free TV/movies? Isn't that a big deal?
Not really. The selection isn't great unless you're a Doctor Who fan. Lots of British stuff. Not much else. The most popular TV shows right now are Doctor Who, Monty Python, Torchwood (a Doctor Who spinoff), and the Terminator TV show. Unless you like British TV (some of it, admittedly, is pretty good), you'll be disappointed by their selection of free videos. You're better off sticking with Netflix. Even if Netflix rates have gone up.
If you're a student, and you buy from Amazon, go for the free year. If you regularly buy from Amazon, consider it. If you buy from Amazon occasionally and you're a huge fan of British TV, it might be worth it. Otherwise, don't bother.