The Testimonium Flavian, a passage from Josephus' Antiquities of the Jews, is often cited by apologists as independent, 1st century confirmation of the existence of Jesus. In particular, Josh McDowell cites this passage in Evidence That Demands a Verdict:
"At that time lived Jesus, a wise man, if he may be called a man; for he performed many wonderful works. He was a teacher of such men as received the truth with pleasure. He drew over to him many Jews and Gentiles. This was the Christ. And when Pilate, at the instigation of the chief men among us, had condemned him to the cross, they who before had conceived an affection for him did not cease to adhere to him. For on the third day he appeared to them alive again, the divine prophets having foretold these and many other wonderful things concerning him. And the sect of christians, so called from him, subsists to this time."
- — Antiquities XVIII, Ch. 3, sec. 3
Although apologists sometimes bring this up in response to claims that there is no independent verification of the existence of Jesus, Josephus was born around 37 CE, which is after the purported life of Jesus. Thus, Josephus was neither a contemporary nor an eyewitness, but was reporting information received from others.
Additionally, this passage is largely considered to be a late Christian forgery or, at best, interpolation.
Why should we suspect that this passage is a forgery? First because, although the church fathers were quite fond of quoting passages which supported Christianity, and though these early church fathers were quite familiar with the works of Josephus, not one of them quotes this passage in defense of Christianity until Eusebius does in the fourth century. We also know Eusebius to be the man who said that lying for the advancement of the church was quite acceptable. He was probably the one who inserted this suspect passage into Josephus' works. Origen, the famous early Christian apologist, even quotes from other parts of Josephus, but somehow neglects to quote our passage. Origen wrote his book Contra Celsus in about 220 A.D.
Secondly, the passage comes in the middle of a collection of stories about calamities- which have befallen the Jews. This would not be a calamity. Thirdly, the passage has Josephus, an Orthodox Jew, saying that Jesus was the Christ. That is a highly unlikely statement for him to have made. The whole passage reads as if it had been written by a Christian. Josephus is made to call the Christian religion "the truth." He would hardly have said that. Although Josephus reports the miracles of a number of other "prophets," he is silent about the miracles attributed to Jesus. Again, this makes no sense when compared to Josephus' known genuine writings. The last phrase in the quotation, ". .. subsists to this time," referring to the Christians, would not make any sense unless it were written quite some time after Jesus had died. Josephus, on the other hand, wrote the Antiquities in about 90 A.D.