Mirkwood is the largest forest in Middle-earth during the Third Age, located east of the River Anduin and north of Gondor and Rohan. The name is probably one of Tolkien's translations from Westron. The Sindarin name is Taur e-Ndaedelos, "Forest of Great Fear." Previous to the building of the evil tower of Dol Guldur it was Greenwood the Great, and after Sauron's defeat, the forest was renamed Eryn Lasgalen, "Wood of Green Leaves."
In Canon Edit
In The Hobbit, Bilbo and the dwarves pass through Mirkwood on their way to the Lonely Mountain. It is dark and full of giant spiders and other nasty things, not to mention an enchanted river that will cause anyone who drinks of it to fall deeply asleep. Despite Gandalf's warning to stay on the path at all costs, the party (especially Bombur the Dwarf) is diverted by lights shining through the trees and the thought of food. The lights belong to some of the Wood-elves of Thranduil's realm, who take exception at the dwarves barging in on their feasting and stirring up the spiders in the process (never mind that the dwarves were actually captured by said spiders and only escaped thanks to Bilbo). The elves capture them and take them before Thranduil, who has no reason to like them and has them locked up in the dungeons of his underground fortress. Bilbo, who was missed due to being invisible at the time, helps the dwarves to escape by means of empty barrels dropped into the River Running for transport back to Laketown.
While all this is going on, Gandalf is busy in Dol Guldur, where an evil being known as the Necromancer had taken up residence and from which come most of the horrible things in Mirkwood. When Gandalf meets up with the dwarves and Bilbo again, he doesn't say much about it except that the Necromancer was driven away; however, in The Lord of the Rings, it is revealed that the Necromancer was in fact Sauron, and he had simply been biding his time there in preparation for the events of the War of the Ring.
Mirkwood doesn't play as much of a part in The Lord of the Rings, and all most readers hear of it is that Legolas, Thranduil's son, comes from there. However, Dol Guldur still stands thanks to Saruman advising against its destruction, and it remains a stronghold of Sauron's during the war. Some of the Nazgûl occupy it and use it as a base of operations for carrying the war to the elves of Mirkwood and Lothlórien (which is why it makes no sense for Lórien Elves to show up at Helm's Deep like they did in the movies).
By the end of the war, the forces of Lady Galadriel and Lord Celeborn overthrow Dol Guldur, and though Thranduil's people have their work cut out for them to clean up the remaining evil things, Mirkwood is eventually cleaned out and renamed.
In Badfic Edit
Mirkwood is one of the most popular choices for a fanfic setting, probably because Legolas lives there for most of his life. However, badfic writers tend to miss certain key points, such as the fact that Mirkwood is a dark, dangerous, and all around Not Nice Place. Legolas does not live in a palace, as many Suethors contend: he lives in a largely-underground fortress designed to keep out things like giant spiders. Also, Mirkwood really isn't the sort of place in which one would want to go out for a romantic walk. The clearings, where they exist, are definitely not full of limpid pools and fantastic flora. If there ARE pools, they'll probably drop you into an enchanted sleep as soon as look at you, and the plants are probably poisonous and/or infested with nasty things.
Despite this, badfic writers will insist on having Legolas or their Sue run off for a bout of angsting into a nice pond, or discovering magical healing berries, or meeting unicorns and fairies. Alas.