Publius Cornelius Tacitus (C.E c.55-c.117) was a senator and a historian of the Roman Empire. His books provide a contemporary history of the Roman Empire beginning with the death of Caesar Augustus in AD 14 and ending with the death of emperor Domitian in AD 96.
His work Annals is important to many Christian scholars, as Tacitus provides an account of the persecution of early Christians by Nero, in which he seemingly confirms some of the events recorded in the gospels:
- A non-Christian, contemporary account confirming the historicity of Christ.
- He confirms the story of Christ's crucifixion at the hands of Pontius Pilate.
The final work of Tacitus was Ab excessu divi Augusti, dubbed Annals, written circa CE 117. It is a collection of sixteen books covering the history of Roman emperors starting at the death of Caesar Augustus (CE 14) and ending with the death of Nero (CE 68). The books cover the reigns of Caesars Tiberius (CE 14-37), Caligula (37-41), Claudius (41-54), and Nero (54-68). The focus is mainly on Tiberius and Nero, as some of the volumes are not extant.
Book 15 of Annals provides this account of the persecution of Christians (Annals, 15.44):
Consequently, to get rid of the report, Nero fastened the guilt and inflicted the most exquisite punishments on a class hated for their disgraceful acts, called Chrestians by the populace. Christ, from whom the name had its origin, suffered the extreme penalty (i.e., Crucifixion) 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 Judea, the first source of the evil, but even in Rome, where all things hideous and shameful from every part of the world find their center 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.
Nero offered his gardens for the spectacle, and was exhibiting a show in the circus, while he mingled with the people in the dress of a charioteer or stood aloft on a car. Hence, even for criminals who deserved extreme and exemplary punishment, there arose a feeling of compassion; for it was not, as it seemed, for the public good, but to glut one man's cruelty, that they were being destroyed.
In this passage, Tacitus is making reference to a fire that burned a large portion of Rome in AD 64. In an attempt to quell the rumor that he had started it, Nero blamed the fire on Christians dissidents, and subsequently arrested, tortured, and executed Christians en masse (though it was clearly Tacitus' opinion that Nero simply disliked Christians; the fire was just an excuse).
Thiessen and Merz, while stating that Tacitus provides few details the source of which is unclear, conclude that there was a Jew named Christus who Pilate had executed, and he began a religious movement which was widespread during Nero's reign. 
Bart D. Ehrman writes, "Tacitus's report confirms what we know from other sources, that Jesus was executed by order of the Roman governor of Judea, Pontius Pilate, sometime during Tiberius's reign." (The New Testament: A Historical Introduction to the Early Christian Writings)
Most of the criticism of the passage in Annals surrounds what the passage proves. There is no scholarly consensus on the issue. The passage is accepted by some scholars as evidence for Jesus' existence, whereas others dismiss it as Tacitus simply repeating the story as Christians told it.
Not an independent source
In The Gospels As Historical Sources For Jesus, The Founder of Christianity R. T. France writes "[there is no] reason to believe that Tacitus bases this on independent information - it is what Christians would be saying in Rome in the early second century." In this and his book Evidence for Jesus, France concludes that Tacitus' reference to Jesus was likely echoing the story being spread by Christians at the time, not something he investigated and sourced independently.
In his book Jesus, Charles Guignebert states that "so long as there is that possibility [that Tacitus was simply repeating the story as it was being told], the passage remains quite worthless." Without more information, which we don't currently have, the passage proves nothing (it can't be used as evidence for or against).
Thiessen and Merz argue that if the source had been an official Roman archive, one would expect him to have referred to Pilate as a prefect instead of as a procurator (he was not a procurator). They thus conclude that the information Tacitus gives about Jesus was not copied from an official source. 
The writings of Tacitus can be considered as either non-Canonical confirmation or as useless, depending on whether the scholar thinks Tacitus exercised due diligence in investigating the story before writing the passage. Given that we are lacking key information, and that the passage itself provides very little detail, a determination about Tacitus' diligence in investigating it cannot be made. Any statement which assumes he did exercise due diligence (i.e. that what he said was based on fact) is speculative.
- The mistake in the use of the title of "procurator" for Pontius Pilatus rather than his actual title of prefect convince many Jesus Myth theory supporters that the passage it at best Tacitus repeating what the Christians claimed.
- It is extremely improbable that a special report found by Tacitus had been sent earlier to Rome and incorporated into the records of the Senate, in regard to the death of a Jewish provincial, Jesus. The execution of a Nazareth carpenter would have been one of the most insignificant events conceivable among the movements of Roman history in those decades; it would have completely disappeared beneath the innumerable executions inflicted by Roman provincial authorities. For it to have been kept in any report would have been a most remarkable instance of chance.
- The phrase "multitudo ingens" which means "a great number" is opposed to all that we know of the spread of the new faith in Rome at the time. A vast multitude in 64 CE.? There were not more than a few thousand Christians 200 years later. The idea of so many just 30 years after his supposed death is just a falsehood.
- The use of the Christians as "living torches," as Tacitus describes, and all the other atrocities that were committed against them, have little title to credence, and suggest an imagination exalted by reading stories of the later Christian martyrs. Death by fire was not a punishment inflicted at Rome in the time of Nero. It is opposed to the moderate principles on which the accused were then dealt with by the State.
- The Roman authorities can have had no reason to inflict special punishment on the new faith. How could the non-initiated Romans know what were the concerns of a comparatively small religious sect, which was connected with Judaism and must have seemed to the impartial observer wholly identical with it.
- Suetonius says that Nero showed the utmost indifference, even contempt in regard to religious sects. Even afterwards the Christians were not persecuted for their faith, but for political reasons, for their contempt of the Roman state and emperor, and as disturbers of the unity and peace of the empire. What reason can Nero have had to proceed against the Christians, hardly distinguishable from the Jews, as a new and criminal sect?
- It is inconceivable that the followers of Jesus formed a community in the city at that time of sufficient importance to attract public attention and the ill-feeling of the people. It isn't the most popular way to convert and bring people into their religion.
- The victims could not have been given to the flames in the gardens of Nero, as Tacitus allegedly said. According to another account by Tacitus these gardens were the refuge of those whose homes had been burned and were full of tents and wooden sheds. Why would he risk burning these by lighting human fires amidst all these shelters?
- According to Tacitus, Nero was in Antium, not Rome, when the fire occurred.
- The blood-curdling story about the frightful orgies of Nero reads like some Christian romance of the Dark Ages and not like Tacitus. Suetonius, while mercilessly condemning the reign of Nero, says that in his public entertainments Nero took particular care that no lives should be sacrificed, "not even those of condemned criminals."
- It is highly unlikely that he mingled with the crowd and feasted his eyes on the ghastly spectacle. Tacitus tells us in his life of Agricola that Nero had crimes committed, but kept his own eyes off them.
- Some authorities allege that the passage in Tacitus could not have been interpolated because his style of writing could not have been copied. But this argument is without merit since there is no "inimitable" style for the clever forger, and the more unususal, distinctive, and peculiar a style is, like that of Tacitus, the easier it is to imitate. Moreover, as far as the historicity of Jesus is concerned we are, perhaps, interested only in one sentence of the passage and that has nothing distinctively Tacitan about it.
- Tacitus is assumed to have written this about 117 A.D., about 80 years after the death of Jesus, when Christianity was already an organized religion with a settled tradition. The gospels, or at least 3 of them, are supposed to have been in existence. Hence Tacitus might have derived his information about Jesus, if not directly from the gospels, indirectly from them by means of oral tradition. This is the view of Dupuis, who wrote: "Tacitus says what the legend said." In 117 A.D. Tacitus could only know about Christ by what reached him from Christian or intermediate circles. He merely reproduced rumors.
- In no other part of his writings did Tacitus make the least allusion to "Christ" or "Christians." Christus was a very common name, as was Jesus. In fact Jospehus lists about 20 in the time Jesus was supposedly said to have existed.
- Tacitus is also made to say that the Christians took their denomination from Christ which could apply to any of the so-called Christs who were put to death in Judea, including Christ Jesus.
- The expression "Christians" which Tacitus applies to the followers of Jesus, was by no means common in the time of Nero. Not a single Greek or Roman writer of the first century mentions the name. The Christians who called themselves Jessaeans, Nazoraeans, the Elect, the Saints, the Faithful, etc. were universally regarded as Jews. They observed the Mosaic law and the people could not distinguish them from the other Jews. The Greek word Christus (the anointed) for Messiah, and the derivative word, Christian, first came into use under Trajan in the time of Tacitus. Even then, however, the word Christus could not mean Jesus of Nazareth. All the Jews without exception looked forward to a Christus or Messiah. It is, therefore, not clear how the fact of being a "Christian" could, in the time of Nero or of Tacitus, distinguish the followers of Jesus from other believers in a Christus or Messiah. Not one of the gospels applies the name Christians to the followers of Jesus. It is never used in the New Testament as a description of themselves by the believers in Jesus.
- Most scholars admit that the works of Tacitus have not been preserved with any degree of fidelity.
- This passage which could have served Christian writers better than any other writing of Tacitus, is not quoted by any of the Christian Fathers. It is not quoted by Tertullian, though he often quoted the works of Tacitus. Tertullian's arguments called for the use of this passage with so loud a voice that his omission of it, if it had really existed, amounted to a violent improbability.
- Eusebius in the 4th century cited all the evidence of Christianity obtained from Jewish and pagan sources but makes no mention of Tacitus.
- This passage is not quoted by Clement of Alexandria who at the beginning of the 3rd century set himself entirely to the work of adducing and bringing together all the admissions and recognitions which pagan authors had made of the existence of Christ Jesus or Christians before his time.
- Origen in his controversy with Celsus would undoubtedly have used it had it existed.
- There is no vestige or trace of this passage anywhere in the world before the 15th century. Its use as part of the evidences of the Christian religion is absolutely modern. Although no reference whatever is made to it by any writer or historian, monkish or otherwise, before the 15th century (1468 A.D.), after that time it is quoted or referred to in an endless list of works including by your supposed historian.
- The fidelity of the passage rests entirely upon the fidelity of one individual (first published in a copy of the annals of Tacitus in the year 1468 by Johannes de Spire of Venice who took his imprint of it from a single manuscript) who would have every opportunity and inducement to insert such an interpolation.
- In all the Roman records there was to be found no evidence that Christ was put to death by Pontius Pilate. If genuine, such a sentence would be the most important evidence in pagan literature. How could it have been overlooked for 1360 years?
- Richard Carrier explains that we are actually missing three years in Tacitus, "We are enormously lucky to have Tacitus--only two unrelated Christian monasteries had any interest in preserving his Annals, for example, and neither of them preserved the whole thing, but each less than half of it, nd by shear luck alone, they each preserved a different half. And yet we still have large gaps in it. One of those gaps is the removal of the years 29, 30, and 31 (precisely, the latter part of 29, all of 30, and the earlier part of 31), which is probably the deliberate excision of Christian scribes who were embarrassed by the lack of any mention of Jesus or Gospel events in those years (the years Jesus' ministry, death, and resurrection were widely believed at the time to have occurred). There is otherwise no known explanation for why those three years were removed. The other large gap is the material between the two halves that neither institution preserved. And yet another is the end of the second half, which scribes also chose not to preserve (or lost through negligent care of the manuscript, etc.)."
- Suetonius doesn't mention this event in his histories.
- And lastly, the style of the passage is not consistent with the usually mild and classic language of Tacitus
Implication to Christianity
If we accept that Annals is evidence enough that Jesus existed, the following points are made:
- There was a Jew named Jesus who founded a religious movement
- He was ordered executed by a Roman prefect named Pilate
- His martyrdom is likely what fueled the growth of the movement
However, we have not made the points that:
- The man had magic powers
- The Biblical account is factual
- The Bible is a reliable historical source
- What people believe today is true