It seems weird, how you're supposed to hold the violin bow, doesn't it? However, the bow hold evolved over many many years of musical people refining it more and more. It's pretty important to hold the bow correctly.
There are several reasons to hold the bow correctly, from the first time you play:
It's not just your bow hold, by the way. Your entire arm plays a part.
Here's a quick rundown on what you need to know:
Finally we come to the violin bow hold itself! At first, you'll find you have to adjust it a lot. Your hand is still getting used to holding the bow. You may get hand cramps (pains) at the base of your thumbn and your fingers might get sore. Just take a break them come back and try again!
Each finger has a job in the bow hold. Let's look at each one:
That's not the whole story:
When you're playing, the thumb will push up a little while the point finger pushes down. It starts with a motion like you're turning a door knob to the left. Your pointer finger digs into the stick of the bow and your thumb pushes up. This makes the hair of the bow dig into the string, to make the sound louder.
To put this another way: Hold the bow on a string and, without bowing, rotate your wrist just a little to the left. You'll have to push up on your thumb and down on your pointer. You'll see the bow's stick dip closer to the bow hair.
This only covers how to hold the bow, not how to actually move your arm.
Try to keep any bowing you do straight. That means your bow and the string should make a plus. The bow should always be perpendicular to the strings.
How do you do that when it's 4 inches from your nose? With a mirror! Go and find a mirror. Play in front of the mirror. Make sure you watch yourself play from the side (meaning you can see the bow moving up and down and you see the strings side-to-side).
Here's an image from an old book by Karl Courvoisier named The Technics of Violin Playing. This is the "plus" I was talking about! Note how this player's bow hold changes depending on where the bow is.