In English you do not ask "Do you have a class", but rather "Do you have class", that's why the first sentence should actually be: "Do you have class today?" The second one is just plain wrong in terms of grammar, you could say it's a grammatically wrong version of the first sentence.

They're both correct, but the second question is more typical of British English. Azrael is correct that you can also ask, "Do you have class today?" If you ask "Do you have A class," you're emphasizing that it is just one class, where "Do you have class," could indicate more than one.

Who is asking whom? If I am a teacher, I might say to my colleague: "have you got a class this afternoon? / do you have a class this afternoon?" But if I were a pupil / student, I might ask a fellow pupil or student: "have you got a lesson this afternoon? Do you have a lesson this afternoon?" To me, the expression 'have a class', when said by a pupil, sounds more like an American expression than a British expression. As a speaker of British English, I would say: Are you teaching this afternoon? Have you got a lesson this afternoon? / Do you have a lesson this afternoon? Have you got French this afternoon?