There are a pile of eBook readers for the iPhone, but only one connects to Amazon's book store, and more importantly, your Kindle library.
As a standalone eBook reader, the Kindle app is a workhorse. Linked to your actual Kindle, it's genius. Your progress is tracked between readers, so you're always on the right page, and any book you download here is synced to your Kindle instantly, and vice-versa.
From the App Developers: Kindle for iPhone allows Apple iPhone and iPod touch owners to read Kindle books using a simple, easy-to-use interface. You can shop for hundreds of thousands of books at www.amazon.com/kindlestore, and wirelessly transfer the books to your iPhone or iPod touch. Enjoy Amazon's low prices on Kindle books, including New York Times Best Sellers and most new releases for $9.99, unless marked otherwise.