There are two ways to fuse from two saiyans into one that's really strong.
The first fusion method is called: the Fusion Dance. Both of the saiyans have to
perform this dance and do exactly the same as the other. Another requirement is that
they have the exact same powerlevel, build and size. Though not all fusions are
succesful, an example is Fat Gogeta. They stay as long as 30 minutes fused and after
that they just turn back into their normal selves. The other type of fusion can only
happen when both of the persons have an earring in the opposite ear. These
earrings are called: the Potara Earrings. Also on this fusion they have to be the
same size, build and powerlevel. This fusion is said to be permanent and well sort
of is, but only when nothing abnormal happens. Also the beings from Planet Namek,
the Namekians, can use a form of fusion. This is done by placing the hand on the
other Namekian. No further things or moves are needed, the mind and power
of the being who the hand was put onto will go into the other. Though it will only
happen when both of the fighters approve. They can turn back into themselves again
anytime they would want to.
Below is the list of fusions I have seen on the show.


Goku and Vegeta turn into Gogeta using the potara earrings.


Goku and Vegeta turn into Vegetto when they have performed the fusion dance.


Goten and Trunks turn into Gotenks also when they have performed the fusion dance.


Supreme Kai and Kibito turn into KaioShin when using the potara earrings.


Piccolo and Kami fuse into Kamiccolo in the Namekian way.

Fat Gogeta

Skinny Gogeta

Fat Gotenks

Skinny Gotenks

The last four fusions are defective ones.