I think in a very, very, broad sense, the word professional means that you get paid to do the work you do. When we do educational programming, we define it as such -- when we ask the kids what a being a professional means, the correct answer is "they get paid." (this is in reference to the dancers). I am a professional stage manager. Joe Schmoe at my bakery is a professional barista. I work for a professional company, where other people get paid to be dancers, marketing directors, etc.
My behavior must be professional, because I work for a professional company. I am expected to act in a certain manner, simply because I am getting paid to do a job. When we cast students in company productions, they are expected to act professional -- i.e., they are expected to act as if they are getting paid to be there. Now, people don't always adhere to this, obviously!
When I do work for community theatres, nobody is getting paid (or getting paid much), but we are all still expected to act professionally, because we are working for a company where people are paying to see the show, and where other staff members are paid to make sure that you are doing your job.
As djemily says, in the theatre world, I think of it more as an attitude and manner than anything else. I am professional, because I do my job well and I treat people well. Other people, even though they're getting paid, are not professional, because they are rude, use bad language in front of the little kids in Nutcracker, talk back to the directors, etc.