In the main Pokémon games you capture and train creatures to build your ultimate fighting team, and when certain conditions are met, they change into more powerful forms, called 'evolution' in-game.
A few have the rather unusual condition that you have to find someone to trade them with, and as a result of the trading process they will evolve into their more powerful forms.
So, you had to track down someone who had a copy of the game and negotiate a trade where you likely traded monsters, let them evolve, and then traded back so you could have your original creature, now powered up.
But! There's a better way!
Pokémon Diamond and Pearl have introduced the ability to trade over the Nintendo Wi-Fi connection. You put up something and what you'll take for it and then wait for someone else to take your offer, kind of like a Pokémon trading post. If what you put up there doesn't get taken, then you can withdraw your request and try again.
But! What if you put up one of the Pokémon that evolve via trading? Then request something that's impossible to get via normal means (such as a Level 1 Blastoise) to ensure that nobody will actually be able to trade with you? Then immediately take your original Pokémon back?
It evolves! Just like you had traded it with someone.
And you have, sort of. You've traded with yourself.