Short answers, pulled from the story.
The First Triumvirate was an informal political alliance between Gnaeus Pompeius Magnus, Marcus Licinius Crassus, and Gaius Julius Caesar, formed around 60 BC. It had no official standing and was not a formal magistracy. The term itself is a modern invention, first attested in 1681 and not widely used until the 18th century.
Each of the three men had a political goal blocked by rivals in the senate. Pompey needed ratification of his eastern peace settlements and land for his veterans; Crassus needed his equestrian clients' tax contracts renegotiated; and Caesar needed political support to pass legislation as consul and win a profitable military command.
In Caesar's consulship of 59 BC, the alliance passed an agrarian law resettling Pompey's veterans, ratified Pompey's eastern settlements, secured a one-third reduction of Crassus' clients' tax contracts, and assigned Caesar a five-year command in Gaul. In 55 BC, after renewing their pact at Luca, Pompey and Crassus won a joint consulship and received major provincial commands in Spain and Syria.
The alliance effectively ended when Crassus died at the Battle of Carrhae in 53 BC, removing the third man who had balanced the other two. Pompey's search for new aristocratic allies after Crassus' death, combined with eroding trust between Caesar and Pompey through 50 BC, led to civil war in January 49 BC.
Modern scholars avoid the term because it implies an official structure that never existed. Harriet Flower prefers "alliance" and "Big Three"; Robert Morstein-Marx in his 2021 book Julius Caesar and the Roman People abandoned the term entirely, arguing the phrase almost inevitably implies a conspiracy theory about the alliance's origins. Mary Beard uses "Gang of Three" in her 2015 book SPQR.
Cicero described it as "three men" exercising a regnum. The satirist Marcus Terentius Varro called it a "three-headed monster." Later historians Suetonius and Livy used the terms societas or conspiratio. The three allies themselves would presumably have called it amicitia, a friendship.