Josephus was a Jewish and Roman historian in the 1st Century AD who wrote (amongst other things) about the Roman-Jewish war that lead to the destruction of the Temple in Jerusalem. In the 18th Century this book was widely read by Christians as it appears to provide historical evidence for Jesus; and Josephus was held up as one of the great historians. However to Jews he was a much more controversial figure and wasn't read or referred to until much later in the Enlightenment. Talking about Josephus's life, times and legacy on In Our Time were Tessa Rajak (University of Reading), Philip Alexander (University of Manchester) and Martin Goodman (University of Oxford and the Oxford Centre for Hebrew and Jewish Studies).
They started the programme with some context for the Jewish-Roman war. In the 2nd & 1st Century BC under the rulership of the Maccabees Judah had become independent. During this period it had formed a sense of itself as a Jewish nation, and so when it became a province in the Roman Empire Judah didn't assimilate into the Empire as well as the Romans would've liked. To some extent the province had a special status - they had a bit more independence than was generally the case. The Jews & their religion were well treated and the Herods ruled as client kings of the Romans. However there was a strand of thought within Judean society that they should be independent, and this was particularly pronounced in the priestly classes and the elite.
Josephus was born in 37AD to a family in Jerusalem who were members of the priestly elite. He was highly intelligent and well educated. Stories about his education have parallels to the stories told about Jesus's education - the bright boy who quickly surpasses his teachers in knowledge and understanding of the scriptures. When the Jewish-Roman War broke out in 66AD he, along with many other intelligent educated sons of the priestly elite, became a general. He had no experience in leading troops, nor did his fellow generals. Unsurprisingly the war is a disaster for the Jews, and the Romans quickly put down what they see as a rebellion of one of their provinces. However, it's important to remember that most of what we know about this war comes from Josephus. And he wrote about it after the fact when he had become Romanised and for a Roman audience. So his bias is against the Jews.
Josephus doesn't entirely whitewash his own actions in the war when he's writing about it. One of the stories he tells reflects badly (by the standards of his community) on him - possibly he only tells it because it was widely known and so better to put his own spin on it rather than miss it out completely. During the war he was leading troops who were holding out against a siege, but they were losing. The acceptable thing to do in these circumstances was to commit suicide rather than surrender, and this is what the others want to do. Technically it's not suicide - each man is to kill another until there is only one left who will commit suicide. Josephus tries to talk his troops out of this, but fails. Eventually there are only two people left, Josephus and one other, and finally Josephus succeeds in talking this other man into surrendering rather than dying. This failure to pursue the honourable path is one of the things that shaped Josephus's later legacy amongst the Jews.
When he surrenders Josephus is captured by Vespasian and taken to Rome as a slave. He tells Vespasian that he has had a vision that Vespasian will become Emperor - which at the time seems extremely unlikely. However, two years later this comes to pass. This little story needs to be taken with a large pinch of salt as the only sources for the vision and timing of the revelation of said vision are Vespasian and Josephus who both have vested interests in it being true.
Josephus worked for Vespasian as a scholar and interpreter, first as a slave and later as a Roman citizen. He wrote a history of the Jewish-Roman War, which is one of the books that he is remembered for. This was written for a Roman audience, and so it was tailored to please his masters and his potential customers for the book. For instance Josephus justifies his defection to Rome by saying that he believes God has withdrawn his blessing from the Jews and it has passed to the Romans. He does also explain the Jewish side of the war and this theme is taken up again in a later book about Jewish history, laws and customs. This is again written with his Roman audience in mind, and is a thorough explanation of his home culture to the people of his new culture.
Josephus's legacy is two-fold. Amongst early & medieval Christians he was revered as a historian, in large part because there is a passage in the Jewish history book which refers to Jesus. This would be the earliest historical (i.e. non-Biblical) reference to Jesus and was tremendously important to Christian readers of his books. The experts all agreed that this reference was almost certainly inserted into the text in the 3rd Century AD by a Christian bishop. It's possible that there was some stub of a reference to Jesus but not the longer description and reference to his Christian followers that is in the version that we now know. The originals of his works did fall into obscurity but in the 18th Century were rediscovered and re-translated. And at that time his history books were widely read by ordinary Christians.
His legacy amongst the Jews was much less positive. He was remembered as a traitor - both for failing to commit suicide when he should've and for later becoming a Roman citizen (and for his belief that God had changed his mind about who His chosen people were). As a result his books were not much read by the Jews, and were not translated into Hebrew. However much later, in the Enlightenment, there was a shift in attitude to the story of Judaism in some parts of the Jewish community. Some wanted their history told in the new scientific style of the Enlightenment era, which was quite a change from the Rabbinical tradition (which is fairly ahistorical). Josephus's works are a good source for what Judaism was like before the Temple was destroyed. They also provide perspective on the immediate impact of the Temple's destruction - as at the time it was assumed it was a temporary setback, not the permanent disaster hindsight showed it to be.