The crucifixion was a common method of capital punishment practiced by ancient Rome.
It was basically an execution, humiliation, torture and example rolled into one. Roman towns and cities usually had a public place selected where crucifixions would be carried out. The people would then be able to witness the suffering of the convicted which would hopefully discourage them from committing the same crimes.
Also an actual cross what not always used for crucifixions. Sometimes it was just an ordinary pole. Sometimes the convicted was tied onto it with ropes, sometimes nailed to it.
Rome was comprised of many territories and tribal identities. The Roman empire had to maintain a delicate political balance in order to maintain control of them, they could ill afford to constantly put down rebellions in their many provinces after all. Jesus (assuming he ever even existed), upset the balance of power in Judea by proclaiming himself (or being proclaimed by his followers) to be the son of the Jews god, and by extension, king of the Jews. Naturally the Jews were not very thrilled, and neither were the Romans. So the only logical solution for them was to execute Jesus and make an example of him, thus appeasing the Jews and avoiding a Jewish rebellion and at the same time reasserting the authority of Roman law in Judea.