DIYer to skilled tradesperson
Vitaliy is basically correct. In most states trades such as plumbing, electrical, gas, oil burners, require a licence. The licence is usually broken down as an apprentice, journeyman then finally a master. Times will vary and requirements and tests vary by state. Specialized permits can only be pulled by masters in the trades. GC's pull the general permit for the structure. 
A general contractor is usually someone that has a good working knowledge of several trades and good trustworthy sub-contractors.  Some states require a licence as a GC and proof of contractor liability insurance. I can tell you that fly by night contractors go by the boards quickly, one needs good skills in estimating, carpentry, problem solving, good credit, at least one million $$$ of liability insurance and the ability to communicate clearly with customers and understand exactly what the customer wants and needs. maybe most important is to be able to realistically tell customers what is possible in the budget allowed and never over promise. We all hear stories about unskilled and bad contractors. They may work for a while, but smart consumers will ask for financial and customer references, and a bad rep will put them out of business quickly. Miscalculate a job and you can lose big bucks, do a bad job and have to claim on your insurance and your insurance will be canceled. 

Most GC's I know are pretty smart guys, good business smarts. They keep up on new products and methods. Lots of our suppliers have product seminars, trade shows and you will see the good guys at them learning to give a better product to the consumer.  The most important factor in being a good GC is experience. The more I see, the more complex the problem, the more I watch and learn from my subs, the better.  I have been doing general contracting for about 13 years now, and I still take time to watch shows like TOH, Holmes on Homes, Ask TOH. I find them a great source of new ideas and methods from some excellent teachers. I also enjoy being a pro member on This old House for pros website.  When I get a tough problem, I can turn to other guys for help and good advise.

I know this answer is a bit rambling, but the bottom line to becoming a successful GC is commitment, experience, knowledge, investment in good tools and people, and growing with a fast changing market.