While you read
Kindle's in-book experience is as straightforward as it gets. To turn the page, simply swipe or tap somewhere in the margin. A tap to the center of the page brings up a status bar along the bottom with location information and a slider if you want to skip ahead or back in the book. There's also buttons at the top of your screen, in case you want to change text size, brightness, margins, line spacing, or background color. For now, the Kindle app offers only black, white, and sepia (my favorite) background options, but hopefully Amazon will add other textures and colors in the future.
The open book icon at the top of the screen lets you quickly go to the cover, beginning of the book, table of contents, or choose the exact location in the book you want. You'll also find a listing here for Book Extras, which are community-curated factoids (think Wikipedia) that are meant to provide readers with helpful information as they read.
Long-pressing a word while reading brings up a dictionary definition and a few options, including highlight, note, and search Wikipedia. With the latest version of the app, you'll be able to use other third-party dictionary apps (such as a medical dictionary for help reading medical books). These added functions really enhance the reading experience by letting you look up extra information without leaving the book.
As with Kindle on other platforms, Amazon's Whispersync technology makes it so the app can remember where you left off and where you placed your bookmarks, which is convenient. However, it would be nice if we could name each bookmark within the same book to make sections easier to find (such as in textbooks for school) rather than try to remember each assigned location.
Other useful extras
The Kindle app for iOS supports the Send to Kindle feature, which allows you to send files to your account in the cloud and access them from your device. Supported file types include JPEG, DOC, RTF, and PDF.
Often overlooked is Kindle's Instapaper-like "read later" feature. Using a bookmarklet or browser extension on your desktop Web browser, you can actually save articles from the Web and use the Send to Kindle function to read them from your iOS device. While this feature might not be the best way to keep a long list of articles you like, it is great for staying on top of ones that you plan to read very soon.
If you're not keen on having to jump through a couple of hoops to browse and buy books, then you might stick with Apple's iBooks. Otherwise, Kindle for iOS is a perfectly capable e-reader app that has several noteworthy features. Plus, it's the no-brainer choice for those already committed to the Amazon ecosystem.