Ummm... :)
Tough question. There aren't any specific corals you "need" for a reef. There are hundreds of different types of corals. You can have any combination of them in a reef tank.
There are 3 main types of corals:
Softies -- mushrooms, zoas, leathers
LPS -- most of the corals you see that have hard skeletons, but flesh that moves, hammer, frogspawn, torch, bubble, candy canes, blastos, duncans, sun corals, elegance, plate corals
SPS -- the corals that look like hard branches or plates, montipora, acropora, stylophora, pocillopora
SPS are the most challenging group to keep, and require the best lighting and water quality. LPS and softies are relatively easy to keep and are great for someone that is new to keeping corals.