Ore

Ore is a block type in Minecraft. Ores are primarily collected for crafting purposes, such as tools, armor, and more. Redstone ore is used as a type of "wiring", which can be used to create redstone circuits, compasses, clocks and more. Ore can also be combined to create a block of the material's type for iron, gold, diamond, and lapis lazuli. Blocks are hard to make, requiring nine pieces of each material to create, and are mainly recommended for storage purposes or building.

Availability
In Minecraft Beta, the following is the distribution of ores according to the altitude. (Layers are number of blocks above the lowest layer of bedrock.)Since the beta 1.8 patch, the highest layers that Gold, Redstone, Diamond and Lapis can be found on are now two layers lower. The graph on the right shows the ore distribution in 1.8.1. * These points are above the maximum map height, and were extrapolated based on the pattern of (2^n, [2^n]+1 to [2^n]+3, [2^n]+4).

Redstone has the same layer and line-size statistics as Diamond, but is generated 8 times per chunk as opposed to 1.

Ore is created by picking two points a certain distance apart, drawing a line between the points, and filling in any Stone blocks within a certain distance of the line with the ore being done in that round. Source (Section 3.11)

In chunks generated prior to 1.6, ore also varies depending on the section of the map it is generated in, due to a bug (fixed in 1.6) in the spawning algorithm. In the quadrant south-west of the point (0,0), ore spawns approximately 50% more than in the quadrant north-east of the point (0,0). The quadrants north-west and south-east of the point (0,0) split the difference between the extremes. (See image below) Any chunks generated after 1.6 do not show this disparity as the spawning algorithm has been fixed. The bug fix did not modify any already existing chunks.

Minecraft Classic
In Minecraft Classic, you could only find exposed Coal, Gold and Iron in large amounts, from roughly 10 to 30 in a vein. Lapis Lazuli, Diamond & Redstone are not found because they were added in later updates.

Trivia

 * Iron and Gold are the only ores that drop the actual ore block when mined without the Silk Touch Enchantment. All the other ores drop the usable item when mined.