The term hatha means union through discipline, but in the western world we use the term as more of an umbrella definition for a yoga practice with physical postures. There are several subcategories of hatha yoga but typically this class has a cohesive flow where poses are held for longer amounts of time. This is a great class for beginners because students can start to unify the breath with movements of the body a little more easily, as there is more time to discover each pose.
This branch of yoga was founded by B.K.S Iyengar and is focused on the specific alignment of the body in postures. You will typically find many props used in these classes such as blocks, straps and blankets. This ensures that the poses are entered and exited in a safe way to minimize injury.
These classes are a little harder to find, and are focused on releasing the kundalini energy that resides in the base of the spine. To awake and release this energy breathing exercises are matched with faster paced body positions and might also include mantras, chanting or meditation.
In Sanskrit, ashtanga means eight limb path. This practice, created by Sri. K Pattabhi Jois, of yoga follows specific sequences that can be more physically demanding than other forms of yoga. There are several levels of sequences ranging from standing positions through to advanced series. Ashtanga yoga places importance on unifying movements of the body with the breath.
These classes seem to be offered the most in studios. Vinyasa is a Sanskrit word meaning place in a certain way or flow. Here, it is up to the teacher to create their own sequence of postures. Like Ashtanga, these classes place an importance on matching movement of the body with the breath. Because this is such a general term it is important to to find out as much as you can about the teacher/class design because the amount of variety between Vinyasa classes can be enormous.
Technically these are two different types of yoga but they are different from the others listed above because of their slower pace. In a Yin yoga class you might only do five postures over the course of the hour, increasing flexibility. In Restorative yoga you may do more postures but the focus is just like the name indicates, to restore the body. Both of these types of yoga calm the nervous system and are more gentle on the body.