Learn Russian: Я студент, а папа врач. - I am a student, but dad is a doctor.

а is a "soft but" or "while." It is used to show a contrast between two things that are different, like you being a student and your dad being a doctor. но is used for stronger contradictions, like "I want to sleep, but I have to work."

Could I use и (and) in this sentence instead?

You could, but it changes the feeling. Using и just links two facts together like a list. Using а is the natural way in Russian to contrast two different subjects doing different things (I do this, whereas dad does that).

Why are there no verbs like "am" or "is" in the Russian sentence?

In the present tense, Russian drops the verb "to be." So instead of saying "I am a student," you literally just say "I student" (Я студент) and "dad doctor" (папа врач).

Do I always need a comma before а?

Yes, Russian punctuation rules are very strict about this. You must always place a comma before the conjunctions а and но.