There is more than one way to hold the harmonica, but I'm only going to describe the most used, most basic style, which should work for nearly everyone, no matter what size hands you have. The objective is to be able to get an air tight cup or seal around the back of the harp.
Place the harp between your thumb and index finger on your left hand with the numbers up, low notes on the left. (Left handed players may want to reverse things, but I'd recommend trying it this way if you can.) Your fingers should be toward the back of the harp so they don't interfere with your mouth or lips. Line up the rest of the fingers on your left hand with your index finger.
Okay, saying it another way, take your left hand and hold all your fingers straight and together. Drop your thumb underneath. Put the harmonica between your thumb and index finger, holes facing you (duh!).
Now, the object of your right hand is to get a good air-tight seal at the back of your harp by cupping and sealing with your left hand.
Open your right hand fingers up and together, thumb out and pointing toward your face. Now bend your hand back at the wrist, tucking your elbow in and keeping your arm vertical. Bring the palm of your left and right hands together.
Remember, you are trying to make a tight cup at the back of the harp. You open and close the cup while you play for "wah" effects, and--an advanced technique--for increased volume using hand resonance. For amplified play, the most important factor is a tight hand cup around the harp and the microphone. This cannot be emphasized enough. You have to work at it and practice to develop a good seal, which is the basis for good amplified tone. It's better to have a good cup than a good mic or amp.
Hand sizes vary a lot, so I can't tell you exactly how to position everything. But I think it's safe to say the index finger on your right hand should be able to curl up over the end of the harp. You can play with the fingers of your right hand either open and straight out, or curled up around your left hand and the harp.
Holding the harp is a personal thing, and comfort is an important ingredient. Just make sure you can get a good air tight cup and you'll be fine, however you hold it.