The Ghost is the ultimate enemy in Spelunky. It is not killable, and will kill you instantly if it touches you. Fortunately, it is possible to make it through the entire game without ever seeing one.
Yeah, I think it's about time I left...
Statistics
Health: Infinite (Unkillable)
Damage: Infinite (Instant kill)
Size: Very Large (2x2)
Corpse?: Obviously not
Stunnable?: No
Special: Passes through walls.
Strategy
There are three instances where it is possible for the Ghost to spawn. The first happens when the timer on a level goes over 2 minutes and 30 seconds. At 2 minutes, you're given a warning: "A chill runs up your spine! Let's get out of here!", along with the game's soundtrack progressively decreasing its pitch. At 2 minutes and 30 seconds the Ghost spawns. You have two options at that point: run to the exit and leave.. or die. The Ghost cannot spawn this way on the first level of Area 1, or on the game's final level. It will also not spawn if you have the Hedjet in your inventory.
The second instance in which the Ghost can spawn is on the Restless Dead level. Picking up that level's idol will immediately cause the ghost to spawn.
Finally, destroying your third or higher altar will immediately spawn the Ghost alongside making the level dark.
Special
For the daring and agile, if the ghost passes over gems, it turns them into diamonds of higher value than what you would normally get for picking them up, even if they are in the ground.
Removal
For the less then satisfied customer of the ghost aka time limit it is perfectly possible to remove it from the game by downloading Spelunky Un-Haunted (a recompiled .exe with one block of code commented out). Note that this does not prevent the ghost from appearing via picking up a Crystal Skull idol on the Restless Dead level, nor will it prevent the ghost from appearing if you destroy your third altar, so effectively the only change it makes is removing the time limit in levels by disabling the warning message/music and stopping the ghost's appearance if you stay longer than 2 minutes and 30 seconds in a level, allowing you to explore to your heart's content.