Organic literally means derived from living organisms. Organic chemistry originally was the study of compounds extracted from living organisms and their natural products. People also accepted vitalism, that there must have been some vital force which created natural products. Any chemist will tell you that there is no difference between a plant derived compound, and the identicle synthesized compound. So even though organic compounds do not need a vital force, they can still be distinguished from inorganic compounds. As I said all organic compounds contain carbon, that doesn't mean though that any carbon compound is organic. If the compound is mineral derived rather than from a living organism, it is inorganic.
Heres a page that lists inmorganic compounds and their properties. You will find many compounds which contain carbon.
http://www.kayelaby.npl.co.uk/chemistry/3_2/3_2.html