The First triumvirate is when three people and in this case Pompey the great, Julius Caesar and Crassus form a group so between them they can all achieve what the want. With all their powers combined they were soon to become very powerful. Crassus with his extreme wealth, Pompey with his influence, and Caesar with his consulship in 59. The triumvirate was entirely unofficial and extremely legal. The Senate especially the Optimates did not like this group so they tried everything in their power to break the triumvirate up. Pompey's goals of the triumvirate were to get land for his Veteran soldiers and ratify his Eastern settlements. Caesar wanted to become Consul in 59. Crassus wanted the Asian tax contract to be revised. The Triumvirate was further reinforced with the marriage of Caesar's only daughter Julia to Pompey.
In 60 Pompey enlisted the aid of a tribune (Flavius) in order to produce a bill aimed at getting land for his veteran soldiers. This was blocked by the senate's use of another tribunes veto and the bill was abandoned. Pompey wanted this law passed and his eastern settlement ratified. A reason for the start of the triumvirate was so that Pompey could achieve this. These goals did get achieved by Caesar in his consulship.
After Caesar had his provincial governorship in Spain after his Praeorship of 62 he wanted to stand for consul in 59. The senate especially the Optimates did not allow him to become consul while having a triumph so he formed the triumvirate so that he could have help from Pompey and Crassus with their influence power and money so that he could become consul in 59. When he became consul the senate made life hard for him by making Bibulus the co-consul who was a hard optimate to wreak his consulship. Caesar in the end became consul.
Crassus did not want much, he only wanted benefits for his clients and in particular he supported the claim of the equites for a revision of the Asian tax collection contract.