As an example, if you asked with that, I'd have replied like "I'm a student" and might have also added "I study Chinese and English at university". :)
Sometimes it is also used to say "Hi! What are you doing?" which is equal to "Привет, чем занимаешься?" but more often it is being said like this "Привет, что делаешь?".
So, basically, you ask this question to someone when you want to know how the person you're speaking to makes money and what he does in order to not to simply die of hunger. :)
Or you can use it as to ask about somebodys' hobbies as well as about somebody's job like "Чем ты любишь заниматься в свободное время?" which means "What do you like doing in your spare time?".
Or you can be asked, say at school "Чем занимаются твои родители?" which is equal to "What do your parents do?" in other words, "What job do they do?" or "What kind of work do your parents work at?".