The Zealots originated about AD 6, when the Romans started ruling Judea and demanded that everyone register to pay taxes. A group of radical Pharisees refused to register, and, led by Judas the Galilean, began a series of revolts and acts of violence, culminating in the First Jewish Revolt in 66-70 AD, which was the trigger for the Romans' destruction of the temple. Because of the threat from the Zealots, the Romans ruled Judea all the more harshly and were particularly vigilant at festivals: it's possible that they were expecting a Zealot revolt on the passover festival during which Jesus was crucified.
The Zealots were eagerly awaiting the arrival of the Messiah, and thought that the Last Days, where God would come and kick the Romans' asses and re-establish Israel as a sovereign nation, were near. They weren't afraid to die, and many of them were killed by the Romans, often by crucifixion. Jesus' disciple, Simon, was a Zealot, and throughout the Gospels you can see Jesus engaging with the Zealot's ideas. Sometimes he challenges them: love your enemies he says; if a soldier forces you to carry something one mile; carry it two. My kingdom is not of this world. But there's a definite Zealot tinge in other places: Jesus comes not to bring peace but a sword, and his birth, not the Emperor's, is gospel good news; he is Lord, not Caesar.
Photo credit: Kaptain Kobold on Flickr