It shows the name when I hover over the enemy, but how do I see what they actually do?

You learn about each creature every time you fight them and the information appears in your Cyclopedia (press J to bring up the journal and it's on the menu).  So your first fight against a creature can be pretty hard though you get hints from the party if things are not working.  Check your Cyclopedia after and you should get an idea of what will work better next time.

 

Edited because I didn't read the question properly.  Not sure about debuffs/buffs from effects during battle. 

Edited March 30, 201510 yr by Kiya