People 'fuse' when two people come together to make one stronger being. There's only a few fuses in DBZ, and only three ways to do them-
Probably the most common fuse. The two people do a special dance to come together. Vegeta and Goku do it to become Gogeta, and Trunks and Goten do it to become Gotenks. This fuse lasts 30 minutes.
The two people put on special earrings (one on each person) to fuse. Goku and Vegeta do it to become Vegeto. It's permanent.
Only Nameks do this, and it's permanent. The people hold each other's hands to fuse. Piccolo fuses with Nail, and later with Kami.