Each character has an Overdrive, and each of their overdrives are useful in battle. With the Skill Entrust, you can also give your companions your overdrive allowing for multiple overdrives in quick succession.
Each character will have a basic type of overdrive and a sub-overdrive which they can then choose. Each will have a different effect and each are as useful as the last.
Credit to ianreck who corrected a couple of spelling mistakes
All in all, there are 17 different modes with which the user can increase the overdrive gauge. You must fulfill the requirements before you get them and below is a list of all 17 and how to get them and what they do.