Basically, violin bow rosin is tree sap. You can buy different roses that are produced from diverse trees. (For example, oak, cedar, pine, etc.) I recommend purchasing the pine tree rosin. (Dark green color) In my opinion, (and the opinion of most violinists) it works better and is far superior to any other violin bow rosin out there. Pine tree rosin helps to create a much improved sound.
The horse hair of the bow has very fine hooks not visible to the naked eye all through the hair follicle. Sound is produced when these tiny hooks "pluck" the string. These small hooks need violin bow rosin to keep their "bite". I suggest applying rosin every time you play your instrument.
Before applying the violin bow rosin, make sure you have tightened the bow to the appropriate tension. With the rosin in your left hand and bow in your right, set the hair on the rosin at the base in front of the frog. With light pressure, steadily and evenly move the violin bow rosin along the entire length of the bow hair to the tip. Then move back down to the frog, to the tip again, and so on until you are sure there is enough rosin on the hair.
There are two methods to do this. You can hold the bow still and move the rosin, or of course, the opposite; hold your rosin still as you move the bow. Either way is fine, but I suggest getting in the habit of moving the bow, not the rosin. With the rosin still, it is applied more evenly and is more efficient overall.
Note: Because rosin is being applied to the bow hair, it is going to end up on the violin strings and the wood of the violin as well. After each time you play, it is very important to remove the violin bow rosin from the strings with a cloth. This creates a loud obnoxious noise, but you'll learn to like it! The rosin will cause damage to the strings if it is not cleaned. It is the same concept for the violin itself. The rosin will eat through the stain and varnish of the wood and create a permanent gray spot. So do your violin a favor, and clean it often.
It is very important to tighten and loosen your violin bow hairs using the tension screw at the base of the violin bow. You need to make it tight enough so there is hardly enough room to fit your pinkie finger amid the wood and the bow hair. Each time you are done playing, it is essential to your bow that you relax the hairs to avoid causing damage. If it is left tightened for too long, the bow will deform. The violin bow is made up of over a dozen parts and is put under a large amount of pressure while the bow hairs are tightened. In most cases, bows are made from "Brazil Wood," which is very strong for this very reason. Even the sturdiest wood can warp, so always be aware of that.