There are two playing modes:
Classic is where when you lose a unit, that's it. They are dead/retired (depends on the unit in question) and you can never use them again.
Casual is where, when a unit dies, they are only gone for the battle and are playable again right afterwards. This does not apply to units who must survive the chapter for recruitment.
After you choose your play style, there are four difficulties to choose from:
Normal difficulty is the recommended difficulty for beginners. I played it on my first run, and its not easy if it's your first time.
Hard is a bit more challenging, but not much. After I played Normal twice, this was easy as cake.
Lunatic is the difficulty YOU SHOULD NEVER ATTEMPT ON CLASSIC. Normal, perfect if it's not your first time. Hard, not much different. The jump from Hard to Lunatic is 10x bigger then from Normal to Hard. You can't even grind for levels until Chapter 5 (plus, there's a Prologue). I'm about half way through it at the moment, and never would have gotten so far on Classic. The difficulty does go down after you can level grind, but its still above Hard in that sense.
Lunatic+ Unlocked after defeating Lunatic. I've never played, but I do know that the villains get exclusive abilities.