2 of the most popular apps out there are the Kindle app (by Amazon) and the Nook app (by Barnes and Noble). Kinda funny that the companies that are trying to sell you an ereader are the same companies that are offering you free apps to do the same functionality. Both of these apps are pretty straight forward. They have the built in bookstores that allow you to purchase books, and download free books, plus, you know, read your ebooks.
Along these same lines is an app put together by a little company called Google. Yeah, apparently the big guns of Google decided to jump into the ebook game with an app called Google Books. The same features as the above books, but the interface is a little more simplified. Also you have access to a ton of FREE public domain books.
The final app that I will cover here today is one of my favorites. Aldiko is an ereader that works with epub and pdf files. By far it is the most flexible ereaders, with a ton of settings you can tweak It also (like all the others) it has an ebook store that has a ton of free and paid books. There is also a premium version available too, so if you like it you can get the premium to support the developer.