All organic compounds contain long chains of carbon atoms. Generally, organic chemistry involves carbon-based molecules, while inorganic chemistry is the study of non-carbon containing molecules. However, not all compounds containing carbon are organic.
When early chemists started studying living organisms, they found that nearly all of the structures and substances present were based on carbon atoms. They called such molecules organic because they thought that only living things could create them. These chemists suspected that living things injected something called a life force into such molecules. This line of thought was soon discarded. Modern chemists understand that there's little fundamental difference between organic and inorganic molecules.
Some examples of organic chemicals are methane, ethane, butane and pyruvic acid. Carbon is involved in so many molecules because it's a flexible atom that can bond with up to four other atoms. This makes it possible to build long, complex molecules from carbon atoms.
Living organisms must use inorganic molecules to live. One important inorganic compound all life requires is water. Water is made of two hydrogen atoms, one oxygen atom, and no carbon atom chain, so it's inorganic. Carbon dioxide, which contains one carbon atom and two oxygen atoms, is not organic because it doesn't have a long carbon chain.