The Batcave is comic book hero Batman's underground lair/command center.
Although depictions of the Batcave have been extremely varied across time and different media, certain constants always adhere. The location of the Batcave is somewhere deep below Bruce Wayne's mansion, Wayne Manor. Generally depicted as a vast and fathomless subterranean cavern complex, the Batcave has at least two entrances - a secret entrance in Bruce Wayne's study (typically depicted as behind a bookcase or grandfather clock), and a larger secret entrance for the Batmobile. Another essential feature of the Batcave is that it be home to lots and lots of bats.
Other common features of the Batcave featured in almost all retellings include a supercomputer, a crime lab, a gymnasium, and garages/hangers for various bat-devices and bat-vehicles. Batman also stores various memorabilia from his adventures and misadventures in a large, open part of the cavern, most notably a giant penny, an animatronic tyrannosaurus rex, and the suits of deceased or departed Robins (see Grayson, Dick and Todd, Jason).
Thematically, the Batcave is where Batman retreats in his darkest moments, when he is grieving, brooding, or stymied or stumped by a villain. In these moments, only his faithful butler Alfred or occasionally, his sidekick Robin, will dare to disturb him. Finally Batman will work through his difficulties and emerge from the Batcave to rule the night once more.
In this way, the Batcave is highly analogous to the Fortress of Solitude at the North Pole, where Superman goes to brood.