When you first launch the application, you are presented with your wish list, and the phrase "Get Yourself a Little Something." From this home tab, you can also see your signed-in username. When you tap the search tab, you are able to type in a search string just as you can on the Amazon.com homepage. Search results are featured in a descending list, ordered by relevance. You can see the categories for the search results by tapping the "By Category" button in the top-right.
Once you're on a product page, you are able to see the same information that you would see on the Amazon.com page, except compacted a little more. Tap the item name/price link, and you get a full product description. Tap the customer reviews section, and you can see all of the reviews. When you're ready to purchase, select "Buy Now," or "Add to Cart" at the bottom of the product page. The picture of the product will then "fall" into the shopping cart tab with a nice animation. You can also see a running count of all the items in your cart. If you decide you don't want the item, just tap the edit button and remove it. When you are ready to complete your purchase, select the "Checkout" button in the top-right corner of the cart tab.
Probably the coolest feature of Amazon's application is located in the "Remembers" tab. In this section, you can take a picture with your iPhone's camera and have it sent to Amazon. In a few minutes, you get a link to the Amazon product page for that item (or the item that resembles it most closely). Note that Mike Schramm's favorite iPhone app, Snaptell [iTunes link], has offered the ability to do the exact same thing for multiple shopping sites. The Amazon "Remembers" feature seems nice, and your results are saved on the Amazon site for later reference. In my tests, after taking the picture I received a link to the product page within 1-2 minutes. This definitely could come in handy if you're in a store and want to find out how much an item sells for on Amazon.
Amazon has really created an all-around great shopping experience for the iPhone and iPod touch. You can download this application from the iTunes App Store for free.