What are the two types of screwdrivers?

There are more than two types, but the main categories are flat head and cross/star screwdrivers.

Actually, there are four: flathead or standard, phillips head, hex or Allen head, or Torx or star head. Torx/star head and Allen/hex head fasters are more often than not bolts, differentiated from screws, but there are a number of different styles of actual screws that are manufactured for these two bits as well.