Vinyasa
Ashtanga is Vinyasa Krama yoga. Vinyasa means body movement with the breath: the body moves on the inhale (puraka), stays, then moves on the exhale (rechaka), in an ongoing cycle. Krama means sequence: each asana prepares the body for the next.
Within a single asana the body flows through a count in Sanskrit: ekam, dve, trini, chatvari and so on. Surya Namaskar A, for example, has nine vinyasas. The first is inhaling while raising the arms overhead; the second is exhaling while folding forward.
Sage Vamana says, Vina vinyasa yogena asanadin na karayet: do not do asana without vinyasa.
Tristhana
Tri means three. Tristhana is the method of flowing your attention through three places while you practise: posture, breathing and gaze. They are always practised together, and they work on three levels: the body, the nervous system and the mind.
Asanas purify, strengthen and give flexibility to the body. Breathing is steady and even, with the inhale as long as the exhale. Drishti, the place you look while in the asana, steadies the mind. There are nine:
Bandhas and the internal fire
Mula bandha and uddiyana bandha are the anal and lower abdominal locks. They seal in energy and give lightness and strength. The tradition places the body's fire four inches below the navel, and long, even breaths keep it steady, neither smothered nor blown out of control.