First of all, google(or similar online search thing, wikipedia is kinda good for all round stuff) is your friend when it comes to researching stuff.

Second:
-Yes, they can have seisures. Which is not very nice when half ton animal falls to ground shaking...
- I'm confused why are you asking what epilepsy is when you want to make a character that has it...
- Usually it has something to do with brains, like tumor or infection. But epilepsy on horses is no way near as common as it is on humans.
Generally a horse with epilepsy is, in my oppinion, dangerous to ride since the seizures are unpredictable. It might have nice life as a paddock decoration so to say, but will probably still need medication. If a wild horse would have epilepsy, it probably wouldn't survive very long.