I practice Inyengar Yoga...my first yoga class
ever was Ashtanga and it was really hard. BUT, I was still hooked. The guy I took the work shop is one of the top Ashtanga teachers in the UK (Chris Gladwell in Bristol)...he did Iyengar yoga for 14 years before he went to Ashtanga. He recommended Iyengar to me because I have weak wrists.
Anyway - Iyengar yoga and Ashtanga yoga are both Hatha yoga styles and use the same asanas.
Ashtanga does focus on the breath and you do more sequence poses. Iyengar yoga holds the asanas for longer.
Like in "Salutation to the Sun" in Ashtanga you will repeat the sequence like 500 (kidding!) times, while Iyengar yoga will concentrate on one element of the sequence (like Uttinasana) and hold it for a long time to gain the body memory. We do the sequences in Iyengar as well, but we take apart the sequence one piece at a time and the concentration is more on alignment. THEN we do the sequences.
I believe Vinyasa (please correct me if I am wrong!) is called "Hot Yoga" in the States. You do the asanas in a heated room? I cannot do that method as I have lo-blood pressure and I would more than likely fall over! Vinyasa does the same asanas as other Hatha styles. (again correct me if I am wrong).
As Chris Gladwell told me at the end of my first yoga class "It doesn't matter what method you practice, as long as you practice".
I have never looked back .... I start yoga teacher training next September (2005).