There are currently two tournaments going on as of now.
The first one is a PO Tournament, which is Pokemon Online. There are several questions about it, so just poke around the Meta or PokeBase, not sure which one. We do these Tournaments monthly.
The second one is a Wifi Tournament, using the actual DS Console to battle. I do believe that they will start doing these monthly as well.
The matchups are made with a Random Number Generator, and the entrants must battle each other in order to go onto the next round (that part is kind of obvious, right?)
Anyway, I hope this helped.