Basically, Aeons are summoned creatures. Summoned creatures have appeared in most previous Final Fantasy games. In Final Fantasy X, however, the summons (Aeons), will act as an actual playable character albeit without the help of the normal characters (they will move).
Aeons have their own unique abilities and attacks, aswell as differing fighting styles.
There are a total of ten Aeons (if you regard the Magus Sisters as 3) in Final Fantasy X, some obtainable as part of the storyline, and others from sidequests.
To view more detailed descriptions of the Aeons, please click on the name of the Aeon you wish to view from the list below.