Slime

A slime is a hostile mob that spawns deep underground and in swamp biomes. They also spawn in flat worlds.

Spawning
Slimes spawn in the Overworld in specific chunks below layer 40 regardless of light levels. They can also spawn in swamp biomes between layers 50 and 70 in light levels of 7 or less.

Slimes will not spawn within 24 blocks (spherical) of any player, and will despawn over time if no player is within 32 blocks and instantly if no player is within 128 blocks.

Slimes require two vertical transparent blocks (e.g., air, signs, torches) to spawn in, with an opaque block underneath. The space they spawn in must also be clear of solid obstructions and liquids. Big slimes require a 3×2½×3 space to spawn, small slimes require a 3×2×3 space, and tiny slimes require a 1×2×1 space (or 1×1×1 if the upper block is not opaque).

Slime size is affected by regional difficulty: chances range from 33% for each size at low difficulty to 16% tiny, 33% small, and 50% big with higher difficulty.

Swamps
In swamps, slimes may spawn at night between the heights of 50 and 70 provided the light level is 7 or less. They spawn most often on a full moon, and never on a new moon.

More precisely, the game checks if the light level is equal to or less than a random integer (from 0 to 7), then if the fraction of the moon that is bright is greater than a random number (from 0 to 1). If these conditions are met and the altitude is acceptable, there is a 50% chance of spawning a slime.

"Slime chunks"
Slimes spawn throughout the world (except mushroom islands) below layer 40 regardless of light level, but only in certain chunks, 1 in 10 of all chunks. These "slime chunks" are determined pseudo-randomly by combining their chunk coordinates with the seed of the world: Random rnd = new Random(seed +                       (long) (xPosition * xPosition * 0x4c1906) +                        (long) (xPosition * 0x5ac0db) +                         (long) (zPosition * zPosition) * 0x4307a7L +                        (long) (zPosition * 0x5f24f) ^ 0x3ad8025f); return rnd.nextInt(10) == 0; That is, using the chunk coordinates to help generate a seed, a random number between 0 and 9 will be generated, and if that number is 0, the chunk will be able to spawn slimes. To convert world coordinates to chunk coordinates, divide by 16 and round down.

Pocket Edition
The slime chunk algorithm in Pocket Edition is different from the PC version. The algorithm doesn't depend on the world seed, thus the chunks slimes can naturally spawn in will be the same for every world. The Pocket Edition slime chunk algorithm was reverse engineered by @protolambda and @jocopa3 and can be found on GitHub.

Drops
Slimes drop experience points equal to their size.

If a slime's size is 1, it will also drop 0-2 slimeballs.

Behavior
Only sizes 1, 2, and 4 spawn naturally. With use of, slimes can potentially range from size 1 to 256 (NBT  tag 0–255).

Slimes move by hopping, which they will do each 10 to 30 ticks ($1⁄2$ to $1 1⁄2$ seconds), and can swim in water. Unlike other mobs, slimes will not stop moving when no players are nearby. Their exact routine is as follows:

The slime will search for a player (or, failing that, an iron golem) within 16 blocks (spherical) distance.
 * If no target is found, they wait 10 to 30 ticks ($1/2$ to $1 1/2$ seconds). Then they will change direction, by a random amount up to 57.26° (1 radian) left or right, jump, and repeat the process.
 * If a target is found, the delay before jumping will be $1⁄3$ as long (3 to 10 ticks), and the slime's direction will be set directly toward the target before jumping.

A slime's health is equal to their size squared, and its dimensions are 0.51 blocks times its size in each dimension. When a slime attacks, it will deal damage equal to its size, except for size 1 (smallest) slimes, which cannot do damage.

A slime's jump distance also depends on their size; a slime will jump a distance slightly farther than its length.

When the player kills a slime and the slime's size is larger than 1, it will die and spawn 2-4 new slimes equivalent to its size divided by 2, rounding down.

A slime's attack speed is twice that of other melee-combat mobs. Mobs like zombies and spiders attack at a rate of one hit per second, while slimes attack at a rate of two hits per second. Slimes damage all players and iron golems they collide with, unlike other mobs that only damage targets they specifically attack.

Data values
Slimes have entity data associated with them that contain various properties of the mob. Their entity ID is.

Pre-1.4 spawning
The (pre-1.4) slime spawning algorithm is discussed in greater detail in this forum topic, but note that the algorithm has changed many times and older information may no longer be accurate.

Trivia

 * Tiny slimes do not prevent you from sleeping.
 * Slimes hop much faster when pursuing a player.
 * From one large slime, the maximum experience a player can get from killing it and all the slimes that split from it is 28 XP; 4 XP from the largest, 8 XP from the 4 it splits into, and 16 XP from the 4 each of those split into.
 * Slimes cannot climb vines or ladders.
 * A named large slime that dies produces smaller slimes with the same name.
 * Along with Armor Stands, Slimes are common in Adventure maps to disable interaction with items/blocks.