Wayang kulit is the most popular one where the puppets are made of animal skin. The puppets are held behind a white cloth which is shined by a light so that the puppets' shadows are cast on it. It is these shadows that the audience actually sees.
Left: A wayang kulit puppet, Right: Wayang kulit performancePhotosource (left-right):
Another form of wayang is wayang golek where the puppets are made of wood. And then there are shows where actual people dress up and wear masks to portray the characters which are then called wayang orang.
If you go to Indonesia, I would recommend watching a wayang kulit show. During the performance, gamelan is used as accompanying music. However, beware that real wayang shows last quite long (hours!).
If you don't get a chance to watch a wayang show, you can always buy the puppets. They are sold in many Indonesian souvenir shops, especially in Java. Recently my sister went to a wayang golek factory and bought this puppet. They make quite beautiful decorations in a room.