Yes, you can read Kindle books on iOS devices using the Kindle app. The app also allows you to read newspapers, magazines, textbooks and PDFs.
The Kindle app automatically syncs the last page you read across all your devices, allows you to buy books directly from the Kindle store, and allows you to read the first chapter of a book for free before your purchase it.
The app requires iOS 6.0 or later, and is compatible with iPhone, iPad, and iPod touch. It is also free to download.