Hatha is a very general term that can encompass many of the physical types of yoga. If a class is described as Hatha style, it is probably going to be slow-paced and gentle and provide a good introduction to the basic yoga poses.
Like Hatha, Vinyasa is a general term that is used to describe many different types of classes. Vinyasa, which means breath-synchronized movement, tends to be a more vigorous style based on the performance of a series of poses called Sun Salutations, in which movement is matched to the breath. A Vinyasa class will typically start with a number of Sun Salutations to warm up the body for more intense stretching that's done at the end of class. Vinyasa is also called Flow, in reference to the continuous movement from one posture the the next.
Ashtanga, which means "eight limbs" in Sanskrit, is a fast-paced, intense style of yoga. A set series of poses is performed, always in the same order. This practice is very physically demanding because of the constant movement from one pose to the next and the emphasis on daily practice. Ashtanga is also the inspiration for what is often called Power Yoga, which is based on the flowing style of Ashtanga with out keeping strictly to the set series.
Based on the teachings of the yogi B.K.S Iyengar, this style of practice is most concerned with bodily alignment. Iyengar practice usually emphasizes holding poses over long periods versus moving quickly from one pose to the next (flow). Also, Iyengar practice encourages the use of props, such as yoga blankets, blocks and straps, in order to bring the body into alignment.
The emphasis in Kundalini is on the breath in conjunction with physical movement, with the purpose of freeing energy in the lower body and allowing it to move upwards. All asana practices make use of controlling the breath, but in Kundalini, the exploration of the effects of the breath (also called prana, meaning energy) on the postures is essential. Kundalini exercises are also called kriyas.
Pioneered by Bikram Choudhury, this style is more generally referred to as Hot Yoga. It is practiced in a 95 to 100 degree room, which allows for a loosening of tight muscles and profuse sweating, which is thought to be cleansing. The Bikram method is a set series of 26 poses, but not all hot classes make use of this series.
Viniyoga is term used by T.K.V. Desikachar to describe the methodology that his father, revered teacher T. Krishnamacharya, developed late in his life. It is based on an individualized approach to each student, creating a practice that suits his or her unique stage of life, health, and needs.