In all four gospel accounts Pilate lobbies for Jesus to be spared his eventual fate of execution, and acquiesces only when the crowd refuses to relent. The gospels describe various attempts by Pilate to avoid responsibility include offering to free Jesus instead of Barabbas, including sending him to Herod Antipas for trial (Luke 23:7) and telling the Jews that he found no fault in Jesus (John 18:38). Only in the Gospel of Matthew does Pilate wash his hands to show that he is not responsible for the execution of Jesus and reluctantly sends him to his death.
Taylor G. Bunch (Behold the Man!, page 158) says washing the hands of all responsibility had long been a Jewish tradition, not unknown among the Romans. Examples of this can be found in Psalms and Deuteronomy.
Psalm 26:6 talks of washing one's hands to show innocence as an accepted custom:
Psalm 26:6: I will wash mine hands in innocency: so will I compass thine altar, O LORD
Deuteronomy 21:1-8 describes a ritual to be followed if a man if found murdered, whereby the people wash their hands and assure God that they were not responsible for the death, asking not to be punished for the crime:
Deuteronomy 21:1-8: If one be found slain in the land which the LORD thy God giveth thee to possess it, lying in the field, and it be not known who hath slain him: Then thy elders and thy judges shall come forth, and they shall measure unto the cities which are round about him that is slain: And it shall be, that the city which is next unto the slain man, even the elders of that city shall take an heifer, which hath not been wrought with, and which hath not drawn in the yoke; And the elders of that city shall bring down the heifer unto a rough valley, which is neither eared nor sown, and shall strike off the heifer's neck there in the valley: And the priests the sons of Levi shall come near; for them the LORD thy God hath chosen to minister unto him, and to bless in the name of the LORD; and by their word shall every controversy and every stroke be tried: And all the elders of that city, that are next unto the slain man, shall wash their hands over the heifer that is beheaded in the valley: And they shall answer and say, Our hands have not shed this blood, neither have our eyes seen it. Be merciful, O LORD, unto thy people Israel, whom thou hast redeemed, and lay not innocent blood unto thy people of Israel's charge. And the blood shall be forgiven them.
The Romans did ritually wash their hands before participating in a temple sacrifice, so it is at least plausible that washing hands could have been seen as absolving guilt in much the same way as the Jewish tradition. However, Pilate was a cruel ruler unlikely to have been concerned about guilt in sending one more man for execution. Wikipedia says Pilate's reluctance to execute Jesus in the gospels has been seen by Anchor Bible Dictionary and critical scholars as reflecting the authors' agenda to place the blame on the Jews, not on Rome.