The competitions happen every so often, but lately they have been more frequent. Basically, when a new competition thread is posted, you can sign up. Hue host will then post a imagineering challenge and you will have to create a submission in a certain amount of time. An example of a challenge might be to create a fifth gate at WDW. Some of the competitions you will work by yourself but some of them you will work with teams. Here are the comps that I know of:
Master of the Parks: Solo
WED99's Think Different Challenge: Solo
One Sentence Contest: Solo
Sorcerer's Apprentice: Teams
Imagineering Tournament: Teams
WEDCar Race: Solo
Spring Training Anyone?: Teams
To get a better feel of them, I would go back and read them because they are really fun to read and it might provide you with some insight.
Upcoming competitions that you can join:
Sorcerer's Apprentice Season 2: Sometime in the summer
Master of the Parks Season 2: I don't know when that's gonna be
I hope that helps.