First implemented in generation 2, pokemon eggs and breeding them are a big part of the pokemon world. Most people breed pokemon for potential shinyness, perfect IVs, or egg moves, or even just to get baby pokemon for their pokedex.
I breed for pokemon shinyness, using the masuda method - instead of a 1/8000 odds of a pokemon being a shiny, if you have 2 pokemon from different game cart regions (say, a pikachu from US and a pikachu from Japan), then the odds of shinyness decrease to 1/~1500, I think.
Here is a shiny venipede I hatched using the masuda method :)
Aren't eggs fun? Especially when you don't know what's inside...wonder what I'll get this time...
(oh c'mon now don't give me that look of "that's not a real pokemon", eggs are an important if not essential part of the games...)