Ashtanga Yoga (ashta anga = eight limbs) is named after the eight limbs (ashta= eight, anga= limb) of sage Patanjali.
Ashtanga Yoga aims to liberate the practitioner from the limitations of the mind, so that she/he can experience one’s true self, purusha or soul that is eternal and blissful.
The first four limbs of Ashtanga Yoga are external practices (yama = restrictions, niyama = observations, asana= yoga posture, pranayama = breath control), and the latter four are internal practices (pratyahara = withdrawal of the senses, dharana = concentration, dhyana = meditation and samadhi = absorption). The last four limbs are internal and unfold gradually as a result of practicing the first four limbs.
The Ashtanga Yoga system that we have learned from Sri K Pattabhi Jois starts and has a strong focus on asana (yoga posture) practice. Asana practice is important because it makes one healthy, strong and stable in body and mind. It is the asana practice that prepares the practitioner for the other, more subtle aspects of yoga.
Mysore practice is named after the city of Mysore in Southern India, where the method was taught by Sri K Pattabhi Jois and is currently taught by his daughter Saraswathi Jois and his grandson R Sharath Jois (see: https://sharathyogacentre.com).
In Mysore style classes, you learn to do the asana practice at your own pace, following the rhythm of your breath. Each practitioner gets individual guidance from the teacher. The asanas are taught one by one, and new asanas are added only when the student can do the previous ones correctly with deep breathing.
The deep breathing and vinyasa (combining the breath and the movement) are essential in the practice. It is the deep breathing that calms the mind and helps the practitioner to find a focused and meditative state during the practice.