The Cave was Batman's secret headquarters located beneath Wayne Manor. The main entrance to that cave was opened by playing three keys on a piano in Bruce's study. It was referred to by Alfred and Bruce as the "foundations" or "southeast corner."
Bruce's ancestors originally used the tunnels that were connected to the cave and transported freed slaves to the North.
Bruce Wayne finds the cave
As a boy, Bruce fell down a well at Wayne Manor where he landed in the heart of that cave. He was then attacked by a swarm of bats and developed a fear of them. Thomas Wayne lowered himself down the well and rescued Bruce from the traumatic event.
After Bruce returned from conquering his fear at the League of Shadows, he noticed that a bat had gotten into his study, which reminded him of the cave underneath the property. He then went back to the heart of the cave with nothing but a robe and flashlight and embraced a massive swarm of bats. In the following days, Bruce returned with Alfred and began to build a secret entrance, storage compartments, and workbenches.
The Batcave was unfurnished with only storage for the Batsuit and the Tumbler. It was a big cave system with a river, three water falls, a ravine, and more. The entrance and exit for the Tumbler was located on a cliff, behind a waterfall. Batman also gave Rachel Dawes the antidote for the Fear Gas in the Batcave when she was on the brink of insanity. Towards the end, when Wayne Manor was on fire, Alfred saved Bruce's life by taking him down to the Batcave in the secret elevator. After Wayne Manor was destroyed, Bruce moved his operations to his warehouse.
After Wayne Manor was rebuilt, the entrance to the Batcave was too. It mostly remained the same as in Batman Begins, but had more equipment including a high-tech super-processing computer. Bruce came to the Batcave in a few occasions in the film as did Alfred. After the "death" of both Bruce and Batman, John Blake inherited the Batcave, which implied that Bruce had intended John to take over the mantle of Batman.