Why is rosin so important? Because rosin is the reason you are able to get any notes out of your violin.
Now about rosin application. Many fiddlers struggle with how much rosin to use. In my experience in being in a lot of settings with other fiddlers, typically too much rosin is used.
My goal with rosining is to end up with the sweetest cleanest and smoothest sound possible. I have been mistakenly called a violinist before (which I was honored to be called) just because I have made a lot of effort trying to meet that goal through the years.
Upon obtaining a new bow, or a freshly re-haired bow, I suggest rosining your bow hair until it grips the strings strongly and you can see some white coating on the strings after playing a little. You don't want to see rosin dust on your instrument until you have been playing for a number of weeks, and then you just want to see it slightly. That is a good indication that you haven't over rosined. Over rosining is a normal problem, so beware! I haven't taken a cleaning cloth to my fiddle top for years, and there is barely any rosin dust on it.
Concerning regular rosining, I recommend waiting to rosin until you feel the bow no longer gripping the strings easily (a kind of slick feeling), as it was after your first rosin application. At these times of applying rosin, do about 25 bow strokes (counting each stroke) with your rosin cake. This should be very sufficient. If you play a lot, you will need to rosin more often, obviously. Again, beware of over rosining!
I have had many students come to their lessons sounding harsh, rough and squeaky. After a few moments of attempting to get off as much rosin as I could, their playing improved radically! Every fiddler should work at getting a beautiful and clean sound on their instruments. Many fiddlers appear like they just have to settle for that famous rough and squeaky sound, but they don't have to. The violin was made to perform better than that, and you can learn to get it to! -Brendan Booher