What determines whether or not an employee is exempt? At my old job (in another state), I was told it had to do with managing others: since I and my co-worker did not manage anyone, we were both non-exempt (he was salaried and I was not). Now, in Illinois, I have no idea if I'm exempt. I am a salaried employee who does not manage anyone. Is exemption based purely on hourly vs. salary, or on the kind of work done?

Strictly based on job duties, although if you are exempt, with a few exceptions such as physician, attorney, certain computer professions) you must be paid a guaranteed weekly salary of at least $455 per week(which does not need to be paid to you on a weekly basis).

The Executive (such as managing thers) exemption is only one classification.
http://www.dol.gov/esa/regs/compliance/whd/fairpay/fs17a_overview.htm