Spawn

Spawning refers to players and mobs (creatures) being created and placed in the game world.

=Player=

When a world is created, so is the player spawn point. Your first view of your map is from this spawn point. The location of the spawn point can be changed if the player digs 3 spaces downwards from it. This will however only move the spawn point a few blocks away from the original location; it is not possible, using this method, to dramatically or controllably alter the location of the spawn point but there is tools that allows you to change the spawn point.

If you die in the game, you will respawn here. Most Tutorials advise marking this point, building a safehouse nearby, or otherwise making it defensible. Items that were in your inventory will be dropped at the place of death. Like all dropped items, they will disappear in 15 minutes if not collected.

In Multiplayer, the spawn point is set by administrators and can even be a Spawn jail.

=Mobs= Mobs spawn no closer than 24 blocks, in a 144x144 area around the player's location after a short delay. A player running through the game world will usually keep ahead of the waves of mobs spawning. Mobs are removed if they become more than 4 chunks away from the player.

Passive mobs
Passive mobs spawn in lit, grassy areas. Torchlight counts.

Enemy Mobs
Enemy mobs spawn in unlit areas on any flat block, however they will not spawn on glass or half Steps.

There are also Mob spawners in dungeons that are a fixed spawn point for enemy mobs.

Minecraft Forum tutorial
Mob Spawning Science by fr0stbyte124.