Crucifixion in the ancient world
Since all the ancient records agree that Jesus died by crucifixion, we need to be clear about the practices to which the term "crucifixion" refers.
The essential element of crucifixion in the ancient world seems to have been the act of attaching the body of the victim to a wood structure: sometimes a tree stump or pole, other times a tree, or even a timber plank. The victim could be alive or dead at the time of impaling, as the core object was not so much to kill the victim as to shame them publicly and to warn other people not to engage in similar unacceptable activities. By the Roman period, the victim was usually alive at the time of crucifixion, so that the process achieved all three objectives: death, humiliation, deterrence.
The extreme humiliation involved in crucifixion is reflected in this excerpt from Cicero's defence of a Roman nobleman and senator at risk of being condemned to crucifixion around 63 BCE:
How grievous a thing it is to be disgraced by a public court; how grievous to suffer a fine, how grievous to suffer banishment; and yet in the midst of any such disaster we retain some degree of liberty. Even if we are threatened with death, we may die free men. But the executioner, the veiling of the head and the very word “cross” should be far removed not only from the person of a Roman citizen but his thoughts, his eyes and his ears. For it is not only the actual occurrence of these things but the very mention of them, that is unworthy of a Roman citizen and a free man. (cited in ABD, 1, 208)
According to the literary sources, crucifixion was an ancient and widespread practice. It seems there were many thousands of victims, including several historical events that involved the mass crucifixion of hundreds or even thousands of people at the same time.
Despite this abundance of literary evidence, we seem to have no visual representations of crucifixion and only a single set of human remains that can be identified as those of a crucified person.
Resources for further study:
For an excellent introduction to crucifixion in antiquity, see the online essay by Joe Zias at the Century One Foundation.
For a more general essay on crucifixion, see the article in Anchor Bible Dictionary (vol 1, pp. 207-210).
For a discussion of the references to crucifixion in Josephus, see James Tabor's online essay at the Jewish Roman World of Jesus web site.