Rosin is the reason that we are able to produce sound on the violin, creating friction that allows the bow to grip the strings. Without rosin, the bow will glide over the stings as if sliding on ice. This article provides a guide for how to apply rosin depending on whether the bow isn't gripping the strings at all, has just been re-haired, has areas that do not play evenly, or, is currently playing to your liking.
If the bow has just been re-haired, it may have been rosined by the shop before it was returned to you (using either a cake or rosin powder), or it may not have any rosin on it at all. If the bow is not playing at all when it is returned to you, you will want to use short targeted strokes against the cake of rosin, to slowly work the rosin into the hair, for the entire length of the bow. This may take a bit of time (which is why the shop will often use a powdered version, which can be applied much more quickly).
The same technique can be used if your bow does not seem to be playing at all and is sliding on the strings, even when it has not just been re-haired.
On the other hand, if the bow is already rosined and plays to your liking, then you can just use a couple of swipes, the entire length of the bow.
If there are just one or two spots on the hair that you notice don't play evenly or don't play at all, you can use short strokes with the hair on the rosin just to target those specific spots. However, if the reason for these slippery areas is that the hair has become dirty or oily (particularly at the frog, where your hand may rub against it frequently), applying rosin usually will not correct this problem and the bow may need to be re-haired or at least cleaned by the violin shop.
It is important to be very careful not to use to much rosin. This is a common problem that can result in a scratchy sound. You should not see any white powder coming off the strings and bow when you play. Rosin also tends to build up on strings (especially when one has used too much) and therefore it is important to wipe them down after you play.
Is acceptable to rosin in both directions for bowed strings (except for bass).
