Originally Posted by Shykat
I don't need to read on my computer (PC).
Then no, you're not searching for the same thing as me. I'm only talking about syncing between a Mac and an iPhone.
The Kindle app will do what you want. You can upload your own e-books as "personal documents", for example by using the "send to kindle" app on your computer. You can convert books in epub format reasonably easily to the mobi format for Kindle, using Calibre etc.
Also iBooks might do the same thing, but I really don't know - I only have one iOS device, so I've never tried syncing my own epubs. Probably there's another topic covering that...