A fusion is where two characters combine powers, either by wearing the Potara Earrings or by doing the fusion dance. If you are an alien race (e.g. Kaioshin or Namek), fusions are done much differently. This way, characters can act as one and become stronger. These are some of the fusions:

Gotenks
Goten and Trunks fused by the fusion dance in order to beat Majin Buu in DBZ. Goku taught them the fusion dance, but it took them many tries in order to get Gotenks to be perfect. This fusion was capable of doing many different attacks than Goten and Trunks could do (e.g. Super Ghost Kamikazee Attack) and they can even reach the SS3 level. They did not successfully defeat Buu because they were very cocky and Buu still overpowered them.

Gogeta
In DBGT, when Vegeta and Goku were fighting against the Black Star Dragons, SS4 Goku and Vegeta decided to fuse by the fusion dance in order to beat Omega Shenron. However, this fusion does last much longer than when Goku and Vegeta fused into Vegeto. Gogeta has more of Vegeta's attributes. When Gogeta attempted to defeat Omega Shenron, he was almost successful. He was doing really well, but unfortunately, he used too much power and the fusion broke up. Soon, weakened Omega Shenron was defeated.

Vegeto
While still trying to defeat Majin Buu, Goku and Vegeta fused while using the Potara Earrings. Goku was actually going to fuse with Gohan, but Gohan was absorbed by Buu, so he needed to find someone else. He was going to fuse with Hercule, but it dawned on him that his last and probably best choice would be Vegeta. This fusion was supposed to last forever, but since Super Buu absorbed them, the fusion wore off and Goku and Vegeta decided to destroy the earrings (they crushed them in their hands). Unfortunately with this fusion, Buu was not defeated.

Namek Fusions
Piccolo fused with Nail and Kami (at different times). Nail and Piccolo fuse in order to defeat Frieza (Piccolo was the end result because Nail and all of his powers were absorbed into him). In DB, Piccolo also fused with Kami.

Old Kai
This fusion between one of the Kais and an old witch was a total accident. The Potara Earrings caused them to fuse. This Kai isn't really old. He just looks old because of the witch.

Kibitoshin
This fusion is bascally just another accident with the Potara Earrings between Kibito and Supreme Kai.