Brief Summary:
There are two types of fusion in Dragonball, one is the Fusion Dance, a temporary fusion that requires the participants do perform a dance. The dance must be symmetrical to each other or the fusion wouldn't work. Another type of fusion is the use of the Potarra Earrings. This is a permanant fusion, and the result is hundreds of times stronger than the Fusion Dance. There is no risk of a failed fusion because all there is required is the earring to be worn on opposing ears by the participants.

Goku learned the fusion dance during the 7 seven he was dead in between the Cell game and the 25th Budoukai. This fusion is to have to have to people perform a dance, they have to be mirror images of each other. This will result in a new warrior who has the attributes of the two warriors who performed the dance. The new warrior will be significantly stronger if the dance is performed correctly.

Requirements/Conditions:
The two fighters performing the dance must be similar in size, build and power, their power levels have to be almost exactly the same, this isn't a problem because most of the Z fighters can control their power levels and lower their power to match the weaker one's.
The dance must be performed correctly for a successful fusion. Any mistake can result in a defective fusion. In movie 12 Goku and Vegeta fused wrong and became a fat Gogeta.
The fusion will last for 30 minutes, at the end of the 30 minutes both fighters will return to their original selves, there is no way to end a fusion before the 30 minute time limit.
After a fusion ends, both fighter must have a resting period of one hour before the dance can be performed again.
Note: Goku taught Goten and Trunks the fusion dance when Goku came back to Earth for one day. Goten and Trunks can fusion together to become Gotenks. Goku and Vegeta can also fusion, they fusion to become Gogeta.

This is the fusion that was given to Goku by Kaioshin, this type of fusion is hundreds of times better than the Fusion Dance. In order for fusion to take place, two people need to wear the Kaioshin earrings.

Requirements/Conditions:
A set of Kaioshin Earrings
The two fighters who are involved in the fusion must wear the earrings on opposite ears.
Each person can only have one potarra earrings fusion in a lifetime.
The effects of the fusion is supposed to be permanent.
Note: Goku and Vegeta used the Potarra Earrings during the Buu saga to become Vegetto. They were released from the fusion when Vegetto was absorbed by Buu.

Merging:
Besides fusion, there was another type of fusion, one that only Nameks could do, this was called Merging, two Nameks will merge together and become one new powerful being.

The two Nameks will permanently merge together, all that is required is to have the two touching each other.

The new being will retain the form of one of the Nameks involved. In this case, Piccolo stayed the same when he merged with Nail and Kami.

Complete chart of the character Fusions and The resulting Character:

Goku + Vegeta (Fusion Dance) = Gogeta
Goku + Vegeta (Potarra Earrings) = Vegetto
Trunks + Goten (Fusion Dance) = Gotenks
Piccolo + Nail (Merge) = Piconail (Remains Piccolo)
Piccolo + Kami (Merge) = Kamicollo (Remains Piccolo)
Kaioshin + Kibito (Potarra Earrings) = Kaiobit (Remians Kaioshin)
Rou Kaioshin + Old Witch (Porarra Earrings) = Rou Kaioshin