Structure

Every specific group of blocks purposefully formed in Minecraft based on coding is part of a natural structure.

The Overworld
The Overworld contains 16 generated structures.

Trees
Trees are structures created both by the automatic generation of the map and by players (they grow from saplings). They are made of wood and leaves. Trees can range from small trees, often found in groups, to large trees that may require several minutes to harvest. In Beta 1.2, birch trees and pine trees may be found, and in update 1.2(not beta), jungle trees may be found.

Huge Mushrooms
Huge Mushrooms are structures that only naturally occur in mushroom biomes. They are composed of mushroom stalks and either red or brown colored blocks for the cap. They can be grown outside Mushroom Islands by fertilizing red or brown mushroom with bone meal.

Hills
Hills are randomly generated pieces of land in the map. Like stairs, hills are always traversable to their lowest point by virtue of the algorithm which generates them; there is always a place on each level from where the next level can be accessed, meaning that the player can climb a hill one level at a time until they reach the top.

Floating "Islands"
Floating "islands" are structures that float in midair that are not connected to the ground, the sea, hills or cliffs. Floating "islands" are normally just random pieces of floating dirt and stone found near cliffs, but on rare occasions they can be large, floating structures that even have springs and trees on them. Floating Islands are usually found in the mountain and Mushroom biomes.

Springs
Springs are randomly generated blocks of either lava or water that act as a source of their respective material. While both can be found on the vertical side of stone blocks above the surface, lava springs are more often found underground beneath layer 32 in caverns. Both of these sources can be collected and moved via the use of buckets. Springs can only be found in versions from Infdev onward, as fluids behaved differently in older versions.

Lakes
Lakes are small bodies of water in which the player is able to swim, and are a large collection of water spring blocks. Lakes can now be generated above sea level or inside caverns, and are small bodies of water surrounded simply by dirt. When in a winter biome, these small lakes are never initially frozen but can turn to ice. These lakes can also be composed of lava; however, lakes of lava are more rare. Lava lakes found at the surface will be surrounded by gravel, stone, and coal ore.

It is possible for lava lakes to spawn 1 block above bedrock, this deletes the bedrock and replaces it with stone (which on survival could allow access to the void). The best way to find one of these is to go on creative, dig down to the void, then fly across in any direction until one finds stone, then dig up.Note that these are rare so one could by flying for a while.

Rivers
Rivers are long strands of water that cut through or seperates biomes. They have no current. Rivers have also been known to be a reliable source of clay.

Beaches
Beaches usually generate next to oceans and cover all nearby shorelines. They used to come in two varieties: sand beach and gravel beach. Sandstone is located below sand in sand beaches. Gravel beaches had no such border and thus pose dangers like falling into caverns located right underneath them.

Historical notes: Due to the changes in the terrain generation algorithm in the Beta 1.8 beaches were removed completely from the game. However, as of Snapshot 12w01a and update 1.2, sand beaches have made a return, but the way they look and generate are not the same as before. Gravel beaches did not appear at all in 1.1. After Snapshot 12w07a, the generation of beaches has been greatly improved.

Caverns
Caverns are caves and tunnels that are automatically generated under the ground in various places. Caverns are composed mainly of stone, and contain small amounts of ores. For more details, see the Caverns article.

Mineral Veins
A Mineral Vein is a natural deposit of ores. Players can come across these veins in caverns or anywhere where there is natural stone. There are six kinds of ores, from common to rarest: Coal, iron, Redstone, lapis lazuli, Gold, and diamond.

Ravines
Ravines are tall, long cracks of air, about 5-20 blocks in height. The width is barely a few blocks, usually no more than 5, sometimes making ravines hard to navigate from below. Ravines can have small ledges along the top, which are possible to travel along. Ravines can be found at any level, and also scarcely appear on the world surface, forming canyons. Such canyons do, from time to time, appear underwater where safe entry is possible.

Ravines can connect to caves, dungeons, abandoned mines, and any other generated structures. Due to the large surface area of their walls, may also have water/lava flowing down them (either from a spring or an underground lake of the lava/water).

NPC Villages
NPC Villages are generated on flat lands and are a site for NPC Villagers. They are composed of several different houses, farms and often a smithy and/or a tower. Sometimes the blacksmith of the village may have a chest with some useful items in it, such as diamonds or iron tools and armor. The Villagers move about in the area around the village. They are the only structure to generate on the world type Superflat, and being that they require large amounts of flat land, this makes them more abundant.

Desert Wells
First publicly introduced in 1.2.1, these well-like structures built of sandstone blocks and slabs generate only in the desert biome. Although assumed to be wells, their exact purpose and function is yet to be known. According to players, they are a rare sight.

Dungeons
Dungeons are small, mostly underground, one-room spaces bordered by moss stone and cobblestone, and typically contain chests with rare items, and a hostile Monster Spawner in the center, which guards the chests. The chests may contain rare valuables, such as cocoa beans, music discs, saddles and golden apples.

Abandoned Mine Shafts
Abandoned Mine Shafts are structures generated underground that consists of branching mining tunnels with wooden supports, and broken rails passing through it. This is also the only place in Minecraft where venomous Cave Spiders can be found.

Strongholds
Strongholds are structures that contain multiple rooms, doors and other aspects. (Some of these rooms include libraries, fountain rooms, dungeon-like rooms, ect.) Stone brick, mossy stone Brick, cracked stone brick, doors, and iron bars are the materials that make up the strongholds. They also contain Silverfishes, which one should be careful of when digging around in strongholds. Portals to The End are found in specific rooms in strongholds. Chests may be found containing not-very-precious resources such as books, redstone, and compasses.

The Nether
The Nether, though vast, contains 2 generated structures.

Lava Sea
Lava Seas are, as the name suggests, great oceans of lava which are found at and below level 30 in the Nether. They make a large portion of the Nether, and are extremely common. They can stretch for hundreds of meters in any direction, and are usually bordered by netherrack (or more rarely soul sand). In some instances these "seas" have been known to cover an area so vast that attempting to cross is nearly impossible.

Nether Fortresses
Nether Fortresses are very large complexes made mainly of nether bricks. Nether fortresses can often tunnel through netherrack, and the inside of the tunnels will be cleared of netherrack by the terrain generator. You can find Blaze spawners in balconies and nether wart inside.

The End
The End only contains 1 generated structure.

Obsidian Pillars
Obsidian Pillars are structures that generate in The End. They first appeared in 1.0.0 with the introduction of The End. No part of them are in the ground; there is no obsidian below the lowest block of obsidian you can see without digging. Ender Crystals will spawn on top of each one to heal the Enderdragon.

Trivia

 * It has been highly hinted at by Notch and Jeb that desert wells will serve for a boss summoner.
 * This may be related to the two new sandstone textures as well.