According to the myths, the great war will be preceded by three years of harsh winter, during which morality will break down. Then the bonds holding some of the most fearsome monsters will break, and the creatures that have been trying to cause chaos for centuries will finally succeed. The wolves that have been chasing the Sun and Moon will devour them, and the serpent Nidhogg will chew through one of the roots of the World Tree. The monsters will join forces with the frost and fire giants, including Loki, who will escape from his prison. Sources mention two ships, one made of the nails of the dead, that will convey the giants and their allies to the final battleground; as well as the giants invading Asgard by way of the rainbow bridge Bifrost, which would break behind them. The gods will be assisted by the Einherjar, the virtuous dead who had been dwelling at Valhalla. The exact order and location of the events seems to vary from one source to another, but the last battle will be fought on the plain of Vigrid, which is ten thousand square leagues in area, and fills the same basic role that Armageddon does in Christianity. While we know that Armageddon refers to Megiddo in Israel, where many battles have been fought over the centuries, I haven't seen any speculation as to whether Vigrid might be a real place. At some point in the fighting, the wolf Fenrir will kill Odin, who will in turn be avenged by his son Vidar. Thor and the Midgard Serpent Jormungand will take out each other, with the thunder god killing the serpent but falling to the dead snake's venom. Eventually, after many more deaths on both sides, the giant Surtr will end the whole thing by setting all nine worlds on fire with his magic sword.
As is generally the case with mythology, this death of everything is followed by a rebirth. Some of the younger gods will survive the destruction, as will two humans, who will repopulate the world. In the new world, crops will grow by themselves, and peace and happiness will reign. The main dwelling place of the new gods will be Gimli, a hall that shines more brightly than the sun (and the source of the name for the main dwarf in Lord of the Rings). I seem to recall seeing some different takes on the new world from after Christianity started to gain prominence in Scandinavia, in which there's only the one god after the Aesir die. While an interesting way to try to reconcile old and new beliefs, wouldn't that mean Ragnarok would have to have happened already without anyone noticing? ("Hey, did you hear that a giant wolf ate the sun the other day?" "No, I was inside all day.")
The Norse apocalypse has become a popular theme, and the name "Ragnarok" has also been used for things that don't have an immediate connection to the end of the world. Indeed, most of the Google results I found when searching for the term did not refer to the battle, but to the RPG Ragnarok Online. Other references that come to mind are the Esper Ragnarok from Final Fantasy VI, Ragnarok Canyon in Battletoads, and the avian villain Ragna Roc in Piers Anthony's Two to the Fifth.