iBooks can only read PDF and epub e-books.
So, blom98's suggestion of getting Calibre is an excellent idea. It's a great e-book management software in itself (it's what I use to keep track of my books). Plus, it does an excellent job of converting mobi files to epub - what you'll need to do.
I, too, keep my ebooks in Dropbox. In fact, you can set up your Calibre library to be entirely stored in Dropbox. Makes it much easier to access your library from anywhere (using the "Open In..." function).
Good luck.
Marilyn