There are many different school of Yoga, and each of them came up with their own style to differentiate and focus according to their philosophy. In my understanding, each Yoga School took the same basic postures and modified them to fit to their style. For example some schools perform yoga slow paced with gentle flow, some will move from one posture to another forcefully and some even use heat as a tool. If we search the web we can come across more than 40 styles of Yoga, however mainly 4 styles are commonly popular and adopted by most of the yoga practitioner. Following are the major (popularly known) styles of Yoga. Please keep in mind, these are the styles of Hatha Yoga. Majority of Yoga practiced today under different brand name and style are basically form of Hatha Yoga.
1. Hatha - This style of Yoga is foundation of all the Yoga Styles. This knowledge was compiled during 15th century. It incorporates Asanas (postures), Pranayama (regulated breathing), meditation (Dharana & Dhyana) and kundalini (Laya Yoga) into a complete system that can be used to achieve enlightenment or self-realization. The ideal way to practice the Hatha Yoga poses (asanas) is to approach the practice session in a calm, meditative mood. Sit quietly for a few moments, then begin the series, slowly, with control and grace, being inwardly aware as the body performs the various poses selected for the practice session. Do not overdo the asanas or try to compete with others. Take it easy and enjoy.
2. Vinyasa, Ashtanga or Power Yoga - These three styles are basically same, it is built upon Hatha Yoga as core foundation with a flow. It began with Krishnamacharya in early 1900 as Vinyasa, later this knowledge was passed to his disciple Sri K. Pattabhi Jois, given a name to his style as Ashtanga around 1950s. This style of yoga is physically demanding as it involves synchronizing breathing with progressive and continuous series of postures producing intense internal heat and a profuse, purifying sweat that detoxifies muscles and organs. The result is improved circulation, flexibility, stamina, a light and strong body, and a calm mind. Further this style in west is more popularized as Power Yoga. However, there is slight difference among them while each type is a flow class– Vinyasa involves no structured flow, a teacher can choose a new flow in a every class, there are pauses in each flow. Ashtanaga has a definite flow, you follow exact same structure in every class at faster speed, and less pauses between poses. Power Yoga is fast faced Vinayasa.
3. Iyengar or Alignment Based Yoga - This style of yoga was developed by B.K.S. Iyengar more in late 1950s, promotes strength, flexibility, endurance, and balance through coordinated breathing and poses that require precise body alignment. The poses are generally held longer than in other styles of yoga. In Iyengar, you slowly move into a pose, hold it for a minute or so, and then rest for a few breaths before stretching into another. Equipment like cushions, blankets, straps, and blocks to help the less flexible also distinguishes Iyengar from other types of yoga. Although Iyengar incorporates the traditional postures, or asanas, that make up the broader category of hatha yoga, the cushions and other props revolutionized yoga by enabling everyone -- even the elderly, sick, and disabled to practice. He was also a student of Krishnamacharya.
4. Bikram or Hot Yoga - This style of yoga that is a comprehensive workout that includes all the components of fitness: muscular strength, muscular endurance, cardiovascular flexibility and weight loss. The founder, Bikram Choudhury, was a gold medal Olympic weight lifter in 1963 and is a disciple of Bishnu Ghosh, brother of Paramahansa Yogananda, (Autobiography of a Yogi). One of the unusual but most beneficial aspects of Bikram's yoga practice is the 95-105 degree temperature which promotes more flexibility, detoxification, and prevention of injuries. This is the only yoga style that specializes in using the heated environment. This style only uses 26 poses.
There are several different styles taught by other masters and some mixed and matched and gave a new name, some style focuses on one area than other such as meditation, breath or energy work.