I searched much of Google, but sites mostly talk about other types of serves, and the underhand serve only gets a couple of quick lines. Even when they do talk more about the underhand serve, they don't indicate which part of the hand it is best to hit the ball with.
To start by stating the obvious: as soon as you've played a small amount of volleyball, you really want to be learning how to do an overarm serve.
That said, an underhand serve is typically performed with the fist, specifically with the fingers where you've curled them into the palm. The only other thing to note is that you must throw the ball up in the air before hitting it; you cannot use your other hand as a "tee" to hit the ball off. This is covered by Rule 12.4.1:
The ball shall be hit with one hand or any part of the arm after being tossed or released from the hand(s).
I have basically seen three different techniques:
- The palm of the open hand. Here you just try to hit the ball with your open hand slightly bent to make contact from the fingers to the root of your thumb. The main contact point should be the part of the palm where the fingers start.
- The clenched fist. In that case, clench your hand into a fist such that the second phalanges of your fingers form a somewhat level platform with the root of the thumb. Use this platform to contact the ball.
- The thumb-side hand edge. This is a bit of an exotic one, but I have seen that as well, especially with older players. For this technique, hold your fingers straight and then lay and slightly press your thumb to the edge of your palm such that it is parallel to your index finger and forms a narrow platform there to hit the ball. When hitting, hold your hand in a way that the so formed platform points forward while the open palm faces towards your body.
Those are the techniques that I know of. I'm with Philip Kendall when he says that you might want to learn doing an overhand serve, but I also know many people who never learnt that or went back to the underhand server due to shoulder problems for example.