What does the home of an elf look like in The Hobbit?
The homes of elves greatly depend on the location and 'type' of elves living there.
Bilbo visits two elven cities during The Hobbit. He also steals food from the Raft Elves after fleeing the Halls of Tharanduil.
The first is Rivendell. Bilbo describes Elrond's house as:
perfect whether you liked food, or sleep, or work, or story-telling, or singing, or just sitting and thinking best, nor a pleasant mixture of them all. Evil things did not come into that valley. (Chap 2)
He also (Chapter 19) makes reference to feeling very refreshed after sleeping in Rivendell.
Rivendell is then further described in the Fellowship of the Ring trilogy:
Sam led him along several passages and down many steps and out into a high garden above the steep bank of the river. He found his friends sitting in a porch on the side of the house looking east. Shadows had fallen in the valley below, but there was still a light on the faces of the mountains far above. The air was warm. The sound of running and falling water was loud, and the evening was filled with a faint scent of trees and flowers, as if summer still lingered in Elrond's gardens. (Book II, Chap 1)
Bilbo then travels to the Halls of Tharanduil. Here:
In a great hall with pillars hewn out of the living stone sat the Elvenking on a chair of carven wood. (Chap 9)
Unlike Rivendell, Tharanduil's Halls are a cave system - partly underground. This allowed segments to serve as a dungeon - although the normal use of these rooms may have been much less sinister (storage rooms perhaps). There were:
many passages and wide halls, lighter and more wholesome than any goblin-dwelling, and neither so deep nor dangerous. (Chap 8)
The water gate is important to the plot - allowing the dwarves to escape.