Technically, the word "Hatha" refers to any sort of physical yoga practice (as opposed to the yoga of service, devotion, or knowledge). But when you see "Hatha Yoga" on a class schedule, it usually means slow, classical-style yoga. Poses are held for thirty seconds and up, with rests in between each pose to observe the effects.
The most detail-oriented, anatomically precise school of yoga, Iyengar yoga makes extensive use of props in order to allow every body, no matter what condition, to experience the yoga poses. This doesn't mean it's easy; you'll hold challenging poses for several minutes. Developed in the 1930s by BKS Iyengar, a student of Krishnamacharya, it was brought to the U.S. in the 1970s.
A methodology more than a specific school, students of T.K.V. Desikachar (son of Krishnamacharya) tend to use the breath as the barometer and signpost for sequencing. Poses might be assigned based on therapeutic needs.
When you want a nap, try restorative yoga instead. Putting the body in a comfortable position and focusing on the breath triggers the relaxation response (slower/lower heart rate, metabolism, breathing, blood pressure, and brain waves) in as little as five minutes. It's "active relaxation" because the mind is gently focused, and the body is carefully stimulated as well as relaxed.
Bikram Yoga is a set sequence of 26 poses, including two breathing exercises. It's taught in a mirrored room heated to 100° or higher, so you can stretch safely. Founder Bikram Choudhury successfully copyrighted his sequence; spin-offs are generically known as "Hot Yoga."
Vinyasa means "flow." Poses flow from one to the next, without pausing. Time is measured in breaths, not minutes. Some schools teach set sequences; some let the practice evolve in the moment. Classes are usually fairly athletic, providing a full-body workout.
Ashtanga Yoga was first taught in the 1940's by Sri Pattabhi Jois, another student of Krishnamacharya. Each student gradually memorizes an opening sequence, one of six middle sequences, and a closing sequence. They're always done in the same order, with the same number of breaths. Traditional, "Mysore" Ashtanga is self-led: students drop in, and proceed at their own pace.
A fairly broad term, Power Yoga classes are usually based strongly on the Ashtanga sequence. The sequence and breath count might vary; the room might be heated.
Developed by John Friend, this "heart-opening" style of yoga teaches five main principles of alignment: open to grace, muscular energy, inward spiral, outward spiral, and organic energy. Classes are moderately paced, with a solid background in anatomy.
Ana Forrest developed her eponymous yoga to battle her own physical abuse, drug addiction, epilepsy, and bulimia. Long, detailed explorations of each pose push students through and beyond their physical boundaries.
The style of yoga practice known as kundalini was popularized in the 1960s by Yogi Bhajan (founder of the Yogi Tea company). It's focused on awakening the latent energy at the base of the spine. You'll get a good buzz from all the breathing exercises, and improve your nervous, endocrine, and subtle systems through various chants, meditations, and calisthenics.
Sivananda Yoga is a flexible sequence of twelve basic poses, preceded by two breathing exercises and several rounds of sun salutations. It was introduced to the West in the 1960s, and now has ashrams around the world. The philosophy emphasizes five tenets of good health: proper breathing, exercise, relaxation, diet, and positive thinking.
There are many other modern fusions. Try a local blend; see what works for you!
All teachers' photos courtesy of their respective websites. Restorative Yoga image courtesy of Yoga Art & Science, Creative Commons Attribution-NonCommercial-ShareAlike 2.5 License