Why Ashtanga? What Does it mean?
The term "ashtanga" comes from the Yoga Sutra of Patanjali, where it refers to classical yoga's eight (ashta)-limb (anga) practice. (Some yoga scholars such as Georg Feuerstein maintain that Patanjali's real contribution to yoga was kriya yoga, the "yoga of ritual action," and that the eight-limb practice was borrowed from another source.) The eight limbs are restraint, observance, posture, breath control, sense withdrawal, concentration, meditative absorption, and "enstasy." This last word, which means "standing inside of," is Mircea Eliade's translation of samadhi, which literally means to "put together" or "bring into harmony." In samadhi, we "stand inside of" our true Self in preparation for the ultimate state of classical yoga, the eternal "aloneness" (kaivalya) of that Self in the purity and joy of its being.
While Patanjali's underlying dualism between Self and nature has long been out of favor, his eight-limb method still influences many modern schools of yoga. One of those schools is the currently popular Ashtanga Yoga developed by K. Pattabhi Jois from the teachings of T. Krishnamacharya (father of T.K.V. Desikachar, brother-in-law of B.K.S. Iyengar, and mentor to both).
Since I'm not an authority on this practice, I asked Ashtanga teacher Richard Freeman to explain. He replied that the Krishnamacharya-Pattabhi Jois system is indeed modeled on the eight limbs of Patanjali; the emphasis, however, is on the correct performance of the third limb (posture) as a means of realizing all the limbs, including, of course, samadhi. Since we in the West sometimes focus exclusively on posture and overlook the other limbs, Richard believes that Pattabhi Jois calls his system "Ashtanga" in part "to encourage his students to look into the whole practice more deeply" and integrate all the limbs.