Vines

Vines (also called Ivy) are non-solid blocks, $1/16$ of a block in thickness, that can only be placed against a solid block, such as onto walls and ceilings.

Obtaining
Vines can be mined with any tool, or without a tool, but shears are the quickest. If mined without shears, it will drop nothing.

Natural generation
Vines are naturally generated in jungle trees of both sizes will have vines on their trunks and canopy edges. However, player-grown 2x2 jungle trees will only generate vines on their trunks, and 1x1 trees will have no vines at all. Vines are also naturally generated in oak trees in swamplands.

Usage

 * Vines can be climbed, but only when they are next to a solid block. This makes them a cheap alternative to ladders.
 * Vines absorb all fall damage.
 * When descending vines, if the vines overhang an empty space, it is possible to continue descending down the vines as if there were a block there. You can't climb back up the vines without solid blocks behind all of the vine pieces, though.
 * Vines block mobs' line of sight, so it is possible to sneak up onto mobs by hiding behind vines. Because of this, Endermen can safely be viewed through vines. You can also shoot through vines, but skeletons can't.
 * Vines can be used to hide other blocks or passageways such as doors or even portals in a similar way to Paintings. This allows the player to create hidden passageways or concealed blocks.
 * Sneaking on vines causes the player to hang onto them, even if the vines are not next to a block. This can be used to "hover".
 * Breaking a block after vines are placed on it will cause the vines to disappear unless they are still supported by other vines.

History
When the Adventure Update was showcased at PAX, vines hadn't been given a name, and a mouseover would not show a title. This implies that they didn't have an official name at the time.