Vinyasa Yoga
Vinyasa yoga is an umbrella term for yoga with breath-connected, flowing asana sequences. It loosely translates from the Sanskrit to “conscious placement” or “mindful sequencing.” Vinyasa is a modern form of yoga, developed by Krishnamacharya and popularized by his student Pattabhi Jois. Some vinyasa styles are slow and others fast. Both are usually strenuous.
Ashtanga yoga is the father of vinyasa yoga. Ashtanga means “eight limbs” and refers to eight ethical, physical, and mental practices outlined by Patanjali in the Yoga Sutras. Ashtanga practice is a series of vigorous sequenced asanas, practiced with keen attention to breath and gaze. It's usually sweaty.
Mysore-style yoga is Ashtanga taught in the tradition method of Jois. The student learns a specific series of asanas and practices them daily. Instead of a class, students come together in a shala, a yoga school, and practice their individual sequences at their own pace, in the presence of the teacher.
Power yoga evolved from Ashtanga. It broke with the traditional sequencing of Ashtanga and incorporates a larger variety of postures into the sun salutations. Power yoga took on an American-fitness friendly name to market yoga to the masses.
Jivamukti is a vigorous and popular form of vinyasa, with a heavy dose of spirituality.