Yoga was originally a spiritual practice to train the body and mind. Modern yoga is often seen as a way to get the body in shape while also relaxing and re-energizing the mind.
Bikram is a type of hot yoga which is practiced in a heated room where you will go through a series of 26 different poses. It was developed around 30 years ago by Bikram Choudhury.
Vinyasa is a class where you will proceed through a series of poses in a smooth transition aiming to link the movements through breath as it is performed.
Hatha is a fairly general term used to describe yoga that teaches physical poses. Often a good class to attend as an introduction to yoga.