In yoga, the term "asana" refers to the individual yoga postures or poses that the practitioner assumes as part of his or her practice. Therefore, asana yoga refers to the physical practice of yoga, and practicing the various postures to experience the benefits. This term may be used interchangeably with hatha yoga, which also refers to the physical postures when doing yoga; in fact, hatha yoga is more commonly used than "asana yoga". This distinction is required because a yoga practice technically has eight "limbs," which refer to other types of behavior toward oneself and others, states of mind, and other practices that do not include the physical yoga postures.
Many Western yoga practitioners do not make the distinction between a yoga practice, and an asana yoga or hatha yoga practice, though devoted yoga practitioners or yoga instructors will often make sure to practice all of the eight limbs. Within the physical asana yoga, however, there are many different types of practices as well. Vinyasa yoga is one of the most common, and is a flowing style of yoga that is often based around a series of sun salutations and concentration on the breath; it is frequently taught in yoga classes of all levels, but tends to be most beneficial for people of intermediate or advanced levels and physical ability.
A gentler form of yoga is known as Iyengar yoga, in which the asanas are held for longer periods of time to ensure the body is in proper alignment. It can be a good place for beginners. Kundalini is another type of slightly gentler yoga, where deeper concentration is paid to the breath. These types of yoga can be excellent for improving overall strength, balance, and posture, but some people find they are not challenging enough. In that case, there are other options available.
Ashtanga yoga and power yoga are often considered together, and though they are very similar, they are not technically interchangeable. Ashtanga yoga requires practitioners to follow a specific series of poses in rapid sequence. Power yoga sometimes follows this series, and sometimes just tries to make the regular yoga practice more aerobic by having practitioners move quickly from one asana to the next. Bikram yoga, also known as hot yoga, is a type of yoga that takes place in a very warm room to encourage the muscles to become more relaxed, and to encourage sweating. These are a few of the most common types of asana yoga practices, but it is by no means an exhaustive list, and devoted practitioners will likely want to do additional exploration to find their preferred types of yoga.