See da picture...

     

The difference in mapping modes is most obvious if you use a picture with a defined pattern/tile on a 3d object.

Object Space = the material is stuck to the object itself.
If you change the size of the object the texture scales with it in proportion.
If you don't want to worry about material getting displaced when you move something in 3d space this is the one to use.

World Space = there's a giant projector surrounding the scene with the texture projected on the object as a pattern.
The pattern doesn't move. Your object picks up a small part of this pattern.
If you move the object in 3d space (the scene) a different part of the 'world' pattern shows.
World space is the hardest to control/use if you want to keep a defined pattern (like bricks, boards, wallpaper etc.) aligned on an object.

Parametric = uses the spatial (geometric) parameters of the selected object to apply a material/texture like a decal. It's stuck to the object. Usually best with picture textures.
Scaling is easiest to control/see in this mode.
Again, the texture doesn't move in 3d space.
The biggest problem usually occurs when people try to use non-seamless textures or *textures that don't match the proportions of the object*.