When you earn Amazon Reward points, they are stored in your account.
Shop With Points - The easiest way to redeem is to simply use them to pay for your purchases when you check out. If you have enough points, you could use them to pay for the entire purchase. If you do not have enough points, you can use them to partially pay for them. What you get is having to pay less because of your points.
If there are not enough points to pay for the full purchase and there is a balance to be paid, you could either then use gift cards to pay the remainder or simply use your amazon credit card to pay.
So that means you can use points and gift cards together on the same purchase (which can result in lots of savings).
Cash Back - Aside from "shop with amazon", you can also redeem your Amazon points for cash back. The exchange ratio is 100 points for $1 (1% return). Redemption level starts at 5,000 points where you will get $50 in cash back or statement credit.
Gift Cards - You can also exchange them for gift cards. You need a minimum of 2,500 points to redeem for a $25 gift card. There is no website available to show potential applicants their gift card partners. But when I called Chase, they said that the catalog will be mailed to you when you get approved and the representative told me that chances are the the partners will be the same ones from their ultimate rewards program.
Travel Rewards - It turns out that you also redeem your points for travel rewards. You simply have to book your travels through their travel redemption center. I suspect they are using the same agent as their ultimate rewards program.