Ashtanga Yoga is actually Ashtanga Vinyasa Yoga. Ashtanga means 'eight limbed' and Vinyasa means 'to place in a special way'. The eight limbs refers to the eight yogic paths in Patanjali's Yoga Sutras.
Sometimes listed as separate types of classes at a yoga studio, Vinyasa yoga tends to be more fluid, and flowing; linking movement with breath. Vinyasa classes can be slow and gentle or more vigorous depending on the class description.
Ashtanga, also a fluid class, is often a very specific set of poses done in a specific sequence.
Power yoga, is a general phrase used in the West as a catch-all to describe vigorous, fitness-based yoga classes. They tend to flow like a Vinyasa class, but much faster and intense.