Josephus
Josephus was a 1st century Jewish historian born in AD 37 who wrote a comprehensive history of the Jewish people near the end of 1st century. In this book, he recounts the stoning of James, calling him "the brother of Jesus, who was called Christ." This passage is considered by most historians and scholars to be authentic and is not generally in dispute. This an important piece of evidence which tells us that someone name Jesus actually lived in the first century and that some considered Him to be the Christ.
Josephus wrote another passage which is more controversial. He wrote:
About this time came Jesus, a wise man, if indeed it is appropriate to call him a man. For he was a performer of paradoxical feats, a teacher of people who accept the unusual with pleasure, and he won over many of the Jews and also many Greeks. He was the Christ. When Pilate, upon the accusation of the first men amongst us, condemned him to be crucified, those who had formerly loved him did not cease to follow him, for he appeared to them on the third day, living again, as the divine prophets foretold, along with a myriad of other marvellous things concerning him. And the tribe of the Christians, so named after him, has not disappeared to this day
There is still much debate over the authenticity of this passage. The current consensus is that Josephus did write something about Jesus here, but that later edits were made by a follower of Christ. The parts in bold italics are those parts which are commonly believed to be later edits, for there is evidence that Josephus was not a follower of Christ and would not have characterized Him in this way. The description of Jesus as a "wise man" and and "teacher" are more consistent with Josephus' style and vocabulary found elsewhere in his work, and are probably the actual descriptions he used.
Tacitus
The Roman Historian Tacitus wrote of Jesus (whom he refers to as "Christus") and the spread of Christianity throughout Rome in his work Annals, approximately AD 116. He wrote:
Consequently, to get rid of the report, Nero fastened the guilt and inflicted the most exquisite tortures on a class hated for their abominations, called Christians by the populace. Christus, from whom the name had its origin, suffered the extreme penalty during the reign of Tiberius at the hands of one of our procurators, Pontius Pilatus, and a most mischievous superstition, thus checked for the moment, again broke out not only in Judaea, the first source of the evil, but even in Rome, where all things hideous and shameful from every part of the world find their centre and become popular. Accordingly, an arrest was first made of all who pleaded guilty; then, upon their information, an immense multitude was convicted, not so much of the crime of firing the city, as of hatred against mankind. Mockery of every sort was added to their deaths. Covered with the skins of beasts, they were torn by dogs and perished, or were nailed to crosses, or were doomed to the flames and burnt, to serve as a nightly illumination, when daylight had expired
There is a great deal of important information in this passage. First, it confirms the life and death of Jesus in Judea but even more importantly, confirms that his death was by crucifixion. According to Christian scholar Edwin Yamauchi, this is an important piece of evidence because death by crucifixion was the "most ignominious death" and reserved for the lowest and most worthless criminals. By Tacitus' own admission, people continued to followed Jesus despite his ignominious death and were prepared to follow him even to the penalty of their own death. This account of the faithfulness of early Christians by an unsympathetic witness is powerful testimony of the life of Jesus.
Pliny The Younger
We also get an account of the spread of Christianity from a Roman provincial governor named Pliny the Younger in A.D. 112. Though he does not speak of Jesus directly, he does recount that Christians in his province cause trouble because they worship Christ and not the Emperor.
They asserted, however, that the sum and substance of their fault or error had been that they were accustomed to meet on a fixed day before dawn and sing responsively a hymn to Christ as to a god, and to bind themselves by oath, not to some crime, but not to commit fraud, theft, or adultery, not falsify their trust, nor to refuse to return a trust when called upon to do so. When this was over, it was their custom to depart and to assemble again to partake of food—but ordinary and innocent food. Even this, they affirmed, they had ceased to do after my edict by which, in accordance with your instructions, I had forbidden political associations.