Shears

Shears are a tool used to obtain wool from sheep, collect Red Mushrooms from Mooshrooms, and harvest placeable leaf, vine, and tall grass blocks by right and left-clicking, respectively.

Uses
Shears are the most efficient way to acquire wool from sheep (each yielding 2 - 4 blocks when shorn). Punching a sheep will not cause any wool to drop, and killing it will only yield a single block of wool. Sheep can be painted with dye before shearing to efficiently produce colored wool.

Shears can be used to obtain leaf blocks legitimately. Using it on any leaf block will yield a single, placable leaf block. The leaf blocks once placed will not decay when too far away from a wood block as of the 1.8 update.

They can also be used to destroy cobwebs in half a second to yield a piece of string, and serve as a faster way to collect wool.

Since Beta 1.8 they can be used to harvest Vines and Tall Grass, in a similar manner to harvesting leaf blocks.

As of Beta 1.9. they can harvest Red Mushrooms from a Mooshroom and turn it back into a Cow.

History
In Beta 1.7, shears were introduced into the game. They were first mentioned by Jeb on Twitter The tweet revealing Jeb's work on shears was written in Swedish (Jag jobbar på shears nu) which translates to "I'm working on shears now". Shears were originally planned as a way to defuse TNT, but Jeb dropped this shortly after deciding TNT would only detonate with redstone or fire.

Before Beta 1.7, sheep would drop 3 wool blocks at most when hit, but shears can harvest up to 4 wool blocks per sheep. This incentive is part of an upcoming plan to have animals stay in the game longer, like allowing sheep to regrow wool (and therefore, shearing as opposed to punching as a collection method for it).

Trivia

 * Shears are the only tool that has a specific right-click action (shearing sheep) and left-click action (collecting leaves, vines and tall grass), and takes damage from both types of use.
 * Hitting a wool block with shears will break it quickly, but not damage the shears themselves.