"To the right of the player is a Taiga (mostly over the ocean), to the left is either a Forest, or Woods, I can’t remember. In the distance is probably tundra." -Notch
According to Wikipedia, Biomes are climatically and geographically defined as similar climatic conditions on the Earth, such as communities of plants, animals, and soil organisms, and are often referred to as ecosystems.
In Minecraft, from the Halloween Update onwards, this means different areas, with different heights, temperatures, humidity, and foliage colors.
Before the Halloween update, when a map was generated it got a 'theme', i.e. Snowy or Grassy. After this update, a single world will include all themes in a logical fashion.
Demonstration video
Pre-Update Saves
Worlds generated before the update will not be changed. However, any new chunks generated (by the player moving outside of the current world's bounds) will generate biomes, just as The Nether will be generated when you first enter it.
Biome Types
There are 11 Biome types: (Note: these names come from Notch's notes)
Biomes Graph
A border between two grass-based biomes. To the right, the grass is more dull/brown; but to the left, it is greener.
A Nether biome loaded in the normal world, this is due to a glitch in chunk generation. (Note the presence of Hellstone)
Rain Forest
A biome with many trees and lakes. Some trees may also have leaves extending to the ground, covering the entire trunk.
Swamp
A biome with many small pools. Reeds are commonly found.
Seasonal Forest
Forest
A biome with many trees.
Savanna
A biome with mostly flat terrain, almost no trees, and dull colored grass.
Woods
A biome with many trees.
Taiga
Snowy terrain with a few trees.
Desert
A biome consisting mostly of sand and Cacti.
Plains
Just like desert, only with grass in the place of sand.
Tundra
Snowy, barren terrain with no trees.
The Nether
This is not an official biome, as it is not usually found in the normal world, but it does have unique geographical features, making it similar to a biome. In rare occasions, most likely due to glitches, nether biomes can be found in the normal world.
Remaining Work
Remaining work on biomes include variable elevations in certain biomes (flat elevation in desert biomes, etc.) in addition to improving the border between arctic and tropical biomes.
Snow currently does not "fall" Meaning that once snow or ice has melted or been mined, it will not come back.