Qualifications
There is one great university that teaches every carpenter—it's called HandsOn. They have locations at every carpentry site in the world, and there's usually cool music playing.
You can become a Journeyman Carpenter by taking a test and undergoing a few years of apprenticeship. Beyond that level is the Master Carpenter—more expertise, more detailed test, more years of apprenticeship (source). You learn everything from how to pick out wood that isn't knotty to how to bid for a job so that all of your costs and risks are considered.
The biggest qualification, though, is just knowing what you're doing. And don't even think about trying to fake it—it'll take a good general contractor about five minutes to figure out whether or not you know your shizzle from your shim.