Below I describe three basic approaches to pizzicato (plucking the string). For all of these examples, it is important to remember to pull the string to the side and not straight out and away from the fingerboard (unless you want the string to snap against the fingerboard like a Bartok pizzicato). Also, it is good to remember that the tone will change depending on where you pull the string. Plucking the string closer to the bridge will bring out a crisper and less booming sound while playing over the fingerboard will make a deeper, less penetrating sound. How you approach the string should be determined by the music you are playing.
Three Basic Approaches
Number One: One of the most widely used approaches is plucking with the side of your index finger from the second joint to the tip. This is one of the best ways to play jazz bass lines, and it is also employed in the orchestra as well. When playing this way, I usually like to anchor my thumb on the side of the fingerboard. However, there is a school of thought that greatly discourages this because it does not allow the player to use the weight of his/her arm to pluck the string. I like to anchor my thumb because I feel I have more control and stability this way. You should experiment to see what works best for you.
Number Two: Another approach, which is primarily used by orchestral players, is to pluck the string with the fatty part of the tip of either the index or middle finger. When I play this way, I generally do not anchor my thumb. However, once again, you should see what works best for you.
Number Three: The third approach, using the tips of both the index and middle fingers, is used by all bass players, and it is generally employed when playing faster passages. When playing this way you alternate between both fingers quickly. The downside to this is that you cannot play with as much power. Also, it takes a lot of practice to get an even sound out of both fingers. When I employ this technique, I generally anchor my thumb. However, this is not the way everyone does it.
What if I have a bow in my hand?
French Bow: Hold the bow at the frog in the palm of your hand with the hair facing away from your palm. Use either the index finger or your middle finger to pluck. I generally do not anchor my thumb when I play this way.
German Bow: Maintain your bow hold and point the tip of the bow to the ground. Then extend your index finger out and use it to pluck the strings.