There are several ways of doing water changes:
1.) Two containers for the betta are kept next to each other, and the betta lives in one of them. When it is time for a water change, he is netted and placed into the other container, and the first one is then dumped, rinsed, and refilled. The process is repeated for the next water change.
2.) The owner puts the betta in a cup along with some of his old bowl water (bonus if you can suck the betta into the cup as you dip for water - no netting!) while the water in the bowl is dumped, rinsed, and refilled. If treated, the betta can be put back into his clean home as soon as the water reaches room temperature; if not, he can be put back in 24 hours, once the water has 'aged'.
3.) Most of the water from the bowl is carefully poured or siphoned out until the bowl is halfway full, and then clean, aged water is added. This is called a 'partial water change', and can be very effective; however it needs to be done more often than a full water change. I recommend the siphoning method, as pouring usually doesn't get the accumulated gunk on the bottom.