The Last Great Time War is a pivotal event in Doctor Who history. It was a cataclysmic was fought between the Time Lords and the Daleks, with the prize being the control of all reality itself. The conflict built from a long history of Time Lord interference in Dalek affairs, dating back to the classic series episode Genesis of the Daleks.
The war is impossible, and meaningless, to classify in terms of when and how it was fought, since it was a war fought throughout time, and with the changing and manipulation of time as one of its primary weapons. Both sides would re-edit history at will to redo the effects of battles, until the fabric of the universe itself begin to strain. In the war's final events (which, again, is somewhat meaningless), the Daleks were poised to destroy Gallifrey and take control of the universe, which the Time Lords countered with a plan to destroy all of actuality and become purely abstract beings. The Doctor, who until then had been a loyal soldier, used a superweapon called The Moment to destroy almost all of the Daleks, as well as all of his own people. The war was also time locked, meaning that time travel into it is now impossible.
Although the Time Lords had always had a streak of arrogance, the terrible toll of the war made them more bitter and vengeful, and they eventually became close to insane, resurrecting the dead Rassilon to lead them. It was Rassilon who came up with the plan to destroy the entire universe, after he had also unleashed superweapons, including the frightening "Horde of Travesties".
Due to its incredibly complicated and awesome nature, the Last Great Time War was never portrayed directly in Doctor Who, except for a few incidental scenes in the special The End of Time. The war was/is basically beyond human comprehension. The horrors of the war can be demonstrated by the fact that The Doctor mentions such things as "The Nightmare Child" and "The Could've Been King and His Army of Meanwhiles and Neverweres" to impress upon The Master how frightening the war was. Since The Master is an insane psychopath who could gleefully torture and destroy an entire planet, this gives some hint of how terrible it was.
Externally to the story, The Last Great Time War takes place between the classic Doctor Who and the revival Doctor Who. It was meant to clear up the messy and convoluted history of Time Lord society that had become more and more distracting in the final years of the original show. Instead, it presents The Doctor as a tragic, unique character. The psychological scars from The Last Great Time War would be a major focus of the revival series.
After the events of The Day of the Doctor, which depicted at least in part the end of the Last Great Time War, several things about the course of the war are more clear, and the ramifications of its end are very different. How this will change the role of the Last Great Time War overall in Doctor Who mythology remains to be seen.