No a sound card is not required to put together a PC. Sound cards are what the name implies, it is a "card" dedicated to sound processing. The same as a Video card is to video.
Motherboards today have integrated audio, so a sound card is not nessicary. Though sounds cards can be capable of providing better audio quality then the integrated sound circutry found on motherboards. Granted, this boost in audio quality is not going to be noticable if you're using standard earphones or headsets.