Patanjali defined Yoga to be the cessation of mind. The mind being the distorting factor in the perception of reality.
In Patanjali's system, Ashtanga Yoga, there are eight steps to salvation.
Yama and Niyama (Do's and Dont's), Asana (Posture), Pranayama (mastery over the breath), Pratyahara (withdrawing of the senses), Dhaaranaa (establishing a focus of one's concentration or dedication), Dhyana (meditation) and finally Samadhi (the state of Bliss).
Of these eight limbs, asana is a relatively minor one. Patanjali devotes only a single sutra in his long "Yoga Sutras" to Asanas, saying only that one should choose a posture in which one can sit attentively, comfortably and without moving for a long time.
But since this is the most demonstrable part of the Yoga system, and it has visible benefits for one's body and health, it is accorded by far the greatest importance by almost all Yoga practitioners. Doing Yoga has become synonymous with doing Asanas. It is also the easiest limb of the eight. Yama's and Niyama's demand a complete change in one's attitudes, Pranayama is a subtle affair, requiring close guidance by an experienced teacher, and very few are really interested in the later limbs because they require total dedication and surrender.
On the other hand, there is a minor "discipline" of Yoga, called Hatha Yoga, propounded by Swatmarama in his "Hatha Yoga Pradipika". This discipline entails keeping the body still in various different postures by firm determination. "Hatha" means stubbornness.
Perhaps it would be more accurate to call the usual teaching of Yoga as the teaching of Hatha Yoga.
The asanas are common to all the different systems but the method of teaching and learning differs. Most small schools have no distinguishing system and they just teach the students based on the ashram's heritage.
However, there can be considered a few important schools of teaching Yoga, especially for the westerners:
Iyengar yoga is the best for fitness freaks and those who value a scientific approach. It takes a gradual approach to precise postures. It uses props, bricks, pillows, bolsters, stools, chairs, ropes, belts etc. to help the beginner. However, it can also be practiced and learnt under an able teacher without all this furniture. Its proponent, B K S Iyengar is very famous for his books and courses and has spent a lifetime learning about the traditional postures and devising variants. He supervises Yoga courses in his academy in Pune, run by his son and daughter.
Ashtanga Yoga is very energetic and requires a good level of health even to begin. It involves vigorous jumps and control of breath while doing the asanas. K Pattabhi Jois, who is credited with this technique, teaches this form of Hatha Yoga in his home in Mysore with his children. One of the important differences in this form of Yoga is the maintaiance of Bandhas (or blockades) in the body while doing the asanas and of a series of postures at the end of each asana cycle (called Vinyasa).
Sivananda Yoga is the most gentle and accomodating. It is basically teaching asanas and pranayama accompanied by lifestyle changes. Instructions regarding a more balanced and sattvic diet, commandments on one's conduct etc. are important in this system. It is a flexible system and can be tailored to a particular student.
Bihar School of Yoga conducts long certificate courses on Yoga. The books published by them are well-illustrated, easy to understand and inexpensive. With a minimal emphasis on esoteric principles, their books and courses teach the basics of Yoga and Pranayama. They take a gradual approach and the asanas are divided into basic, intermediate and advanced poses. They do advocate lifestyle changes but do not compel anyone.
Viniyoga is taught at Chennai at the Krishnamachari Yoga Mandiram by TKV Desikachar, BKS Iyengar's brother and his assistant teachers. The courses are designed with the individual in mind and are very flexible. It is said that after J Krishnamurti harmed his body by too strenuous a Yoga regime (he was originally taught by BKS Iyengar), he had to be re-taught by TKV.
There are additional schools of Yoga. The Vivekananda Yoga Research Institute near Bangalore teaches month long Yoga Instructor courses. The Yoga instruction is good but one is also given heavy doses of questionable Hindu philosophy. Questioning is not encouraged.
There are many institutes in and around Rishikesh which teach Yoga based on one's aptitude and appetite. Yoga Study Centre in Koyalghati, Rishikesh is managed by one Brahmchari Rudra Dev who teaches according to the Iyengar system. I found him to be impatient and insensitive to individual differences amongst his disciples. But if you can brave his egoistic and hard personality, maybe you can learn some delicate points about various props and postures.
These days one Swami Ramdev teaches Yoga everyday live on TV in the morning and evening. His emphasis is on Yoga for healing. He has his ashram in Haridwar.
If you want to remain healthy, pay close attention to your eating habits. A balanced, bland and healthy diet goes a long way in promoting good health.
In many books on Yoga, various postures are recommended as remedies for various health conditions. But there is no agreement on these remedies. It is best to start with simple standing postures and to build one's own programme of daily practice with sensitivity and intelligence.