Gogeta is the fused form of Goku and Vegeta when they perform the Fusion Dance properly.
(In the Japanese version's English subtitles his name is sometimes spelled Gojita.)
Gogeta only exists in Dragon Ball Z movie number 12 and in Dragon Ball GT as Super Saiyajin 4 Gogeta. Since the fusion only lasts a short time there isn't much to say about him except that his personality is something of a mix between Goku and Vegeta, and that (at least as far as DBZ movie 12 is concerned)
He doesn't toy with his opponents. (DBGT breaks many of the established character behaviors and universal rules of DBZ) There is a different form resembling Gogeta resulting from a botched attempt at fusion. He is essentially a very fat and weak version of Gogeta.
During the Kid Buu Saga of the television series Goku and Vegeta perform fusion using the Earrings of the Elder Kai to become Vegetto: whom is similar in appearance to Gogeta but with subtle differences (for example, Vegetto does not become Super Saiyajin). The behavior is also changed - in that assuming the fusion is permanent, they spend great length toying with Gohan Buu. The effects of the earings are meant to be permanent but since Vegetto entered Buu's body, Vegeta and Goku separated from each other due the fact that Buu's body has happened to make fusions irrelevant in it.
Of all the DBZ and DBGT combinations Gogeta proved to be by far the most powerful of them all, but Vegetto hasn't shown his powers at Super Saiyan 4 (this'll most likely will not be shown ever).
After Vegeta managed to turn into Super Saiyan 4, Vegeta tells Goku to do the fusion. Goku and Vegeta perform the fusion. Gogeta dominates physically Omega Shenron without a problem (the power of Gogeta is extremely superior to the Dragon).
Gogeta defends Earth from a powerful ball of negative energy. Gogeta use Big Bang Kamehameha against Omega Shenron (but the dragon survives the impact). At the time that Gogeta would use the Big Bang Kamehameha (the second time) to finally kill Omega Shenron, the fusion ends. Goegta has spkipped Super Sayian 2 and 3 due to Goku and Vegeta becoming a Super Sayian 4 with their time limit over 10 minutes.Gogeta could become both a Super Sayian 2 and 3 due fans.
Stardust Breaker - Gogeta uses this technique against Janemba. ( Gogeta attacks Janemba With a rainbow-colored sphere of energy).
Magic Materialization (Bushitsu Shuyutsu Gen Majyutsu)- Gogeta is able to materialize things out of thin air. This is seen when gogeta SSJ4 fight Omega Shenron, while toying with Omega, Gogeta materializes a confetti cannon.
Dimension Scream- Since Gotenks SSJ3 and Majin Buu had this ability, We can be sure that Gogeta has this ability to... Reason is because he is much more powerful than the two stated before, Although not shown, this would probably be Gogeta's ultimate technique, Its a universal threat and maybe even multiversal... Because when Buu used this technique, Dende stated that "He could break through one dimensional wall after another, the imbalance could cause a collapse", This implies that it could affect other universes to. Though its primary use is to travel between dimensions.
Gogeta has all the abilities of Goku and Vegeta.
Gogeta's full power and abilities are unknown, But humiliating Omega Shenron implies that his powers are limitless, Omega Shenron was marked as universal threat, It took a universal spirit bomb to kill him, Gogeta is considerably stronger than Omega.
Dragon Ball Z: Fusion Reborn
The twelfth Dragon Ball Z movie.