Thucydides was a Greek historian who lived in the 5th Century BC, and is regarded as a "Father of History" alongside Herodotus - although I confess that before I listened to the In Our Time programme about him I had never heard of him. I think he's been seen as more of a "historian's historian", whereas Herodotus is more of a "popular historian". The programme also told me that Thucydides's work is still important in the field of international relations. The experts who discussed him were Paul Cartledge (Cambridge University), Katherine Harloe (University of Reading) and Neville Morley (University of Bristol).
Thucydides was born around 460BC and a citizen of Athens, not much is known about his life. In fact the only details known about him at all are those found in his book on the Peleponnesian War - which includes that he was a general at a particular early point during the war, and he at least lived through the war. This gives a feel for his age as he must've been a mature adult at the beginning of the war yet still young enough to survive till the end. The Peleponnesian War was a conflict between Athens and Sparta, and their allies, which lasted for 27 years at the end of the 5th Century BC. Thucydides's book clearly contains passages written after the end of the war (as he mentions who won - Sparta), but it was never finished. It also doesn't really mention the role that the Persians played which was important later in the war, the experts speculated that if he'd finished the text he may've revised the existing parts to bring in that thread earlier.
Herodotus and Thucydides were writing very different sorts of history, with different purposes. I think they said that Thucydides was writing his history in reaction to the way that Herodotus wrote his - deliberating doing things the way he thought was "proper". For instance Herodotus is the historian as a story-teller. He doesn't necessarily believe all the stories he writes down, but he tells them because that's what the people he's writing about believe. Thucydides in his introductory section says that he is intending to set down the objective truth about what actually happened. This means that he also rejects supernatural explanations of events. Herodotus is also outward looking - partly by the circumstances of recent history but also because of his interests. The big war that Herodotus talks about is the Greek/Persian war of the early 5th Century BC, and his history is of the world outside Greece. By contrast Thucydides is interested in an intra-Greek conflict and in the history of the Greek world. Even, potentially, to the extent of ignoring the Persian role in the Peloponnesian War (although as I said above he may've revised that later if he'd finished the book).
Of course Thucydides isn't as objective as he would like to present himself, and doesn't stick strictly to the known facts either. In contrast to modern historians he doesn't present his evidence, merely says he examined it and has come to the conclusion that what he writes is what happened. So his biases aren't always clear, but in some cases they are obvious. In particular he generally approves of Pericles, and frequently editorialises about his greatness. He also editorialises about the poor decisions by "the mob" who vote for courses of action that Thucydides feels were wrong. There are also sections of the text that are clearly made up to show how something might have happened. The speeches are a good example of this - as well as Thucydides's chronological dicussions of events there are also sections purporting to be speeches given by various people. Pericles is given many of these. In style they sound like Thucydides rather than different individuals, so they definitely aren't accurate representations of actual speeches. Some might be paraphrases of things that Thucydides witnessed, but others are clearly invented out of whole cloth - accounts of secret meetings on the Spartan side for instance that Thucydides was obviously not present for.
In terms of his legacy and his status as a Father of History Thucydides has had a large impact in the past on how historians approach research and objectivity. But all three experts were in agreement that he wouldn't quite fit in in a modern historical department. Modern history also has commonalities with Herodotus's approach - looking at the history of a people as that people see it is an important aspect of approaching history. However in the field of internal relations and of war theory Thucydides is still hugely influential, and his work is still used in teaching at military academies like West Point. Which seems appropriate as that was his primary interest - how different states (cities, nations etc) interact, and what are the causes that lead to conflict between them. Not the causes they use to justify aggression but the underlying conflicts and tensions that get the relationship to the point where aggression is a next step.