Vinyasa means breathing and movement system. For each movement, there is one breath. For example, in Surya Namskar there are nine vinyasas. The first vinyasa is inhaling while raising your arms over your head, and putting your hands together; the second is exhaling while bending forward, placing your hands next to your feet, etc. In this way all asanas are assigned a certain number of vinyasas.
The purpose of vinyasa is for internal cleansing. Breathing and moving together while performing asanas makes the blood hot, or as Yogacharya Shantaram says, boils the blood. Thick blood is dirty and causes disease in the body. The heat created from yoga cleans the blood and makes it thin, so that it may circulate freely. The combination of the asanas with movement and breath make the blood circulate freely around all the joints, taking away body pains. When there is a lack of circulation, pain occurs. The heated blood also moves through all the internal organs removing impurities and disease, which are brought out of the body by the sweat that occurs during practice.
After the body is purified, it is possible to purify the nervous system, and then the sense organs. These first steps are very difficult and require many years of practice. The sense organs are always looking outside, and the body is always giving into laziness. However, through determination and diligent practice, these can be controlled. After this is accomplished, mind control comes automatically. Vinyasa creates the foundation for this to occur.