The cave of the Three-Eyed Raven is a place located Beyond the Wall. It serves as the abode of the greenseer known as the Three-Eyed Raven and a group of children of the forest who protect it and him.
Above the cave lies an ancient and massive weirwood tree with a face carved on its trunk. The weirwood's roots dig deep into the cave and have merged with the body of the Three-Eyed Raven. The entrance to the cave is protected by the magic of the Children of the Forest and prevents wights from entering it, destroying the undead creatures if they attempt to enter.
Bran was guided to the cave by the Three-Eyed Raven in their shared greensight dreams and when he warged into a weirwood tree that showed him another weirwood tree on a hill, while the raven asked him to look for him in there. Eventually, Bran and his group reach the tree, which is above the cave. However, they are attacked by a group of wights before they can enter. Although Jojen perishes at the hands of a wight and his sister's mercy-killing, Bran, Meera and Hodor are saved by a child of the forest, who quickly leads them into a cave. As the remaining wights try to enter after them, they are blown to pieces, and the child explains that the magic that reanimates them has no power inside. She then leads them through the complex cave system of the root-covered cavern to meet the Three-Eyed Raven.
Bran is educated on the history of Westeros by the three eyed raven through the ability to transiently view parts of the past - including the past of his relatives, friends and the origin of the Night King - by use of the tree root system of the cave and the three eyed raven's guidance.
Bran also discovers that he can impact on the past somewhat by using this ability, as shown by his father seemingly hearing his voice as he calls out him during a visit to the events at the Tower of Joy. However despite this the three eyed raven tells him that the past cannot be changed.
The ability to view the past and possibly current (unclear due to the timing of some events - such as seeing an army of walkers and wights - could be present day) ultimately leads to the cave's demise. After Bran channels into the root system, without the Raven's permission and supervision, he dreams of being present somewhere north of the wall alongside the army of wights and the white walkers. He is noticed by the Night King who consequently touches him, awakening him from his dream through the roots. The Raven informs him that being touched by the Night King whilst in this dream has created a mark, which now allows the White Walkers to pass the magical boundary which previously would have prevented them from attacking the cave.
Within a short period of time, the Walkers and their army arrive and invade the cave, killing the three eyed raven, the children of the forest and the direwolf loyal to Bran inside. Bran and Meera manage to escape whilst Hodor sacrifices himself outside the cave by holding the door they escaped from in order to block the wights from coming after Bran and Meera. Hodor is shown being torn at by ravenous wrights as a flash back reveals that Bran warging into a young Hodor in the past to "hold the door" whilst being threatened in the current time by the pursuing wright horde, is what caused hodor to have a seizure in the past which lead to the mental impairment of being only able to say "hold the door" first, and then "Ho-dor" for the rest of his life. Bran realizes this and is visibly saddened, as he and Meera escape into the distance.
Behind the scenes
The cave of the Three-Eyed Raven set was built by repurposing and completely redecorating the Dragonstone set. Given that Stannis Baratheon and his followers leave for the Wall in the Season 4 finale, the production team didn't need to keep the set as Dragonstone afterwards.
In the books
In the A Song of Ice and Fire novels, the cave of the three-eyed crow appears in the fifth book, though its location is not specified. The Lands of Ice and Fire map collection places it within the Haunted Forest, east of the Fist of the First Men and somewhat near the headwaters of the Antler River. Very loosely, if one draws a line straight east from the Fist of the First Men, and another line straight north from Craster's Keep, the cave is in the general area where the two lines intersect. Logically, it isn't as drastically far north as the distant northern edge of the map or the mountain valley of the Thenns - the crippled Bran Stark and his companions could not have traveled such a vast distance on foot.
The cave is actually a great cavern under a wooded hill, deep within a system of caves. The three-eyed crow is seated on a weirwood throne in the great cavern, by an abyss, attended to by the remaining Children of the Forest.