We'll define (pure) braid groups in 3 different ways: as the fundamental group of configuration spaces, as mapping class groups of the punctured disk, and algebraically via generators and relations. To do this we'll develop some background on configuration spaces, including two "forgetful" maps. We'll also define the integral Burau representation, a map which relates braid groups, mapping class groups, and symplectic groups, and whose properties remain an active area of research.