On one occasion, when the soldiers under his command came to Jerusalem, he made them bring their ensigns with them, upon which were the usual images of the emperor. Roman battle standards were considered idolatrous by the Jews. The ensigns were brought in secretly by night, but their presence was soon discovered. Immediately multitudes of excited Jews rushed to Caesarea to petition him for the removal of the obnoxious ensigns. He ignored them for five days, but the next day he admitted the Jews to hear their complaint. He had them surrounded with soldiers and threatened them with instant death unless they ceased to trouble him with the matter. The Jews then threw themselves to the ground and bared their necks, declaring that they preferred death to the violation of their laws. Pilate, unwilling to kill so many, succumbed and removed the ensigns.
There was no physical evidence of Pilate's existence until 1961, when an excavation of an amphitheatre uncovered a block of limestone with a carved inscription dedicted by Pilate to the Emperor Tiberias. (The painting above was created by Giotto in the early 14th century.)
But there are stories, of course; many stories. It is certain that he was prefect when Christ died. There is no particular reason to disbelieve the gospels which, although written long after Jesus died, present a credible story. Jesus was brought to Pilate by the Jewish judiciary, who don't seem to have quite known what to do with him. Jesus was saying things that they considered blasphemous. The accused him of sedition in front of Pilate, saying he opposed paying taxes and called himself a king. Pilate's question to Jesus was, "Are you the king of the Jews?" Jesus' reply is recorded in Greek as being «σὺ λέγεις» which is generally translated as "It is as you say" (King James version) but literally means "you say". There is lots of debate about what he meant by that, assuming it's an actual quote. Our earliest written source is the Gospel of Mark, which (so far as we know) was first written more than 30 years after the fact.
John, characteristically, provides more colour. According to his gospel Jesus said, "You say rightly that I am a king. For this cause I was born, and for this cause I have come into the world, that I should bear witness to the truth. Everyone who is of the truth hears My voice." Pilate's response is one that resonates strongly with the postmodern sensibility: "What is truth?"
The gospels report that Jewish tradition provided for the release of one prisoner at the time of Passover. He put it to the mob (who seem to have been conveniently waiting outside his window), presenting Jesus to them ("Ecce homo") and giving them a choice between Jesus and a guy named Barrabas. Coached by Jesus' enemies, they demanded the release of Barabbas. Pilate is presented as being reluctant to condemn Jesus, because he couldn't see what all the fuss was about. The final, famous moment arrives when, in the gospel of Matthew, Pilate "took water and washed his hands before the multitude, saying, 'I am innocent of the blood of this just person. You see to it.'"
Naturally the question of what Pilate was actually like, what he himself thought of the situation, how he might regard his posthumous reputation, all these are fertile ground for the imagination. And many imaginations have worked on it. According to Eastern Orthodox tradition, he did everything he could to release Jesus and thus should not be blamed. The Ethiopian Church even recognized him as a saint in the 6th century.
Traditional tales relate that Pilate suffered dramatic career setbacks under Caligula and retired to Vienne in France (Gaul), where he committed suicide. There is not a shred of real evidence of this. Folklore continues the story, however; it is said when he died his body was thrown into the Tiber (nowhere near France) but the river was disturbed by evil spirits and rejected him. So they (who?) took his body to Vienne where they threw it in the Rhone, which also wouldn't have him. Finally it was sunk in the lake at Lausanne or, some say, "a deep and lonely mountain tarn" overlooking Lucerne. Every Good Friday the body re-emerges from the waters and washes it hands.
I cannot resist the opportunity for a reference to Monty Python. YouTube has a segment from Life of Brian, where Pilate asks the mob which prisoner to release, but Pilate, as conceived by Michael Palin, has a speech impediment. Hilarity ensues.
Sources: Wikipedia, King James Bible