In British English, 'ill' is the opposite of 'well'. If you feel ill, it means that you aren't well. If you feel sick, it means that you are nauseous and might vomit. If you say 'I was ill this morning' it means that you didn't feel well; if you say 'I was sick this morning', it means that you vomited.
In American English, the usage is different. American English speakers often use them interchangeably, or use 'sick' to refer to minor short-term illnesses (such as flu or an upset stomach) and 'ill' for longer-term and more serious conditions. I'm sure that an American English speaker can give you more details about this, though.