Proof: Tacitus mentions the Christians and Christ after whom they are named.1
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 atthe 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 evenin Rome, where all things hideous and shameful from every part of the world find their centre and become popular. (The Annals 15:54)2
Refutation: Tacitus does indeed speak about a people known as “Christians” and the idea that they are named after “Christ” who was treated with a harsh penalty during the reign of Pontius Pilate but this cannot be used to prove Jesus existed.
Tacitus was a loyal Roman citizen who wrote very unseemly things about Christians. Simply because he connects the term Christian with Christ does not mean that he believed that a person known as Christ existed. He is simply making a statement about a group of people whom he detests.
Tacitus is simply relaying what he has heard from Christians and is not indicating that he has any source for there being a Christ who existed around the first century CE. In addition, even if Tacitus was using official Roman documents there is also no indication that those documents were anything other than what the Christians said about themselves.
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1Matthew Slick. “Non-Biblical Accounts of New Testament Events and/or People.” carm.org. Christian Apologetics and Research Ministry, n.d. [http://carm.org/non-biblical-accounts-new-testament-events-andor-people]
2Alfred Church & William Brodribb. “The Annals by Tacitus.” mit.edu. The Internet Classic Archive, n.d. [http://classics.mit.edu/Tacitus/annals.11.xv.html]