In a nutshell a professional dancer is someone who is paid money to dance or to teach dancing. It needn't necessarily be their full time job, they may just teach a couple of classes, but if they get money then they are professional.
The sad thing is, that anyone who can dance half a dozen steps can legally set themselves up with their own school and start teaching - legally - they do not have to be qualified. This is why when looking for dance teachers you should always look for a qualified teacher - if they advertise themselves just as professional the chances are they are not qualified.
To be qualified they have to take a series of strict examinations to prove their knowledge. Competitors fall into 2 categories - amateur and professional. If amateur they may get sponsorship (but a pittance in the UK) but will not earn money except in certain circumstances; the professional competitors will earn their money teaching, coaching, demonstrating, but even they will take the necessary exams to become qualified - although very often not until they stop competing and want to start judging.
Hope this ramble hasn't bored you to tears! My apologies if it has.
[BTW - yes, it can be very difficult for someone who is a pro dancer to get a mortgage!!!!!!!]