Is it appropriate to attend a class one is not registered for?
In the American system, this is called "auditing" a course. Auditing means that you are attending the class but are technically only listening (auditing => auditory => listening) to the material and not sitting for exams or handing in homework assignments. You can audit a course formally (which often means that this is listed on your transcript) or informally in which you just sit in on the lectures/discussions. For the former/formal, see your local university guidelines. This answer deals with the latter.

Permission of the instructor in all but the larger lectures is always a good idea unless your university has an open classroom policy (as with many European universities). The instructor may have different rules. For example:


In larger lectures, I do not mind auditors (of any ilk) in lecture but they may not attend discussion sections since those are for the paying students.
I allow graduate students to audit my graduate seminars but with three rules: a) They must attend all of the sessions; b) they must do all of the readings; c) they must not be a 'dead body' but must participate in the seminar discussion.


Note that the university itself may have rules on auditing which the faculty member may ignore (for example, I do not mind if local community residents audit my lectures even if they are not registered as auditors through the university).

This is different from PASS/FAIL (also known as CREDIT/D/FAIL) which is another alternative in which you take the class but are allowed a lower level of participation.