Gogeta is a fusion of Goku and Vegeta, created when they successfully perform the Fusion Dance. Gogeta, like all perfect fusions, is extremely powerful, but also very cocky and prone to waste time from his 30-minute fusion time limit.
Gogeta was first formed in order to combat the threat of Janemba. The first attempt at fusion was imperfect (thanks to Vegeta), creating a fat, weak form of Gogeta, which Janemba easily thrashed. The second was the true Gogeta, who, instantly in Super Saiyan 2 form, was easily able to destroy Janemba and restore the citizen of the Other World who was his host. Gogeta fissioned a half hour later, and Goku and Vegeta went their separate ways again.
Dragon Ball GT
Gogeta was formed by Super Saiyan 4 Goku and Vegeta to defeat their ultimate foe, Ii-Shenron. Super Saiyan 4 Gogeta was awesomely powerful, able to humiliate the Evil Shenron with ease. However, unknown to Gogeta, his fusion would last only 20 minutes instead of 30, due to the power involved. When he finally took the situation seriously and began using his Big Bang Kamehameha attack, it was too late- Gogeta broke back apart into his components, who were too weakened to recombine.
The imperfect version of Gogeta created during the battle with Janemba is dubbed "Veku" by some fans.