What does “auditing a course” mean? I am visiting an American university for a year. In my invitation letter and further instructions, it is written that I can audit any course that I want (subject to the agreement of its professor).

Since it is not common in my country, I have no idea what this auditing process is. What should I do?

Is it an official process with official report, or do I simply attend the class like students?
It usually means that you can sit in the lectures, but that none of your work will be graded/marked and you won't get any credit for it. Some universities have official forms that record the fact that you audited a course. Others just do it through personal consultation with the instructor. 

During my first year of grad school, I audited a codes and cryptography course my senior-year undergrad friends were taking. I sat through the crypto part and then bailed on the codes. It was nice. I know a little about RSA and Elliptic Curves now. I got to hangout with my friends and I didn't have to do any work. I think I'm a better educated person for it, but it doesn't really affect my career.