Organic chemistry is the study of organic compounds, a "compound" being a grouping of at least two different atoms and "organic" referring to living things, which must contain carbon and hydrogen. An organic chemist works on things like food, flavoring, petroleum, preservatives, fragrances, plastics and medicine.
Inorganic chemistry is the study of compounds that contain either one or no carbon atoms, which are mostly things that are not alive. This chemistry branch covers a wide range of areas including metals and other materials and the development of analytical instruments, such as those for studying light waves.
Physical chemistry is the study of energy within matter. It involves using principles of mathematics and physics in order to study chemical reaction rates; how light interacts with matter; how electrons, which are part of an atom, move within matter; and how stable or reactive a given substance or chemical process is.
Analytical chemistry analyzes substances in matter. It requires identifying the substance -- qualitative analysis -- and how much of it is present in the matter -- quantitative analysis. For example, in deciding whether a white powder is sugar or cocaine, an analytical chemist would be called in. The analytical chemist is the type of chemist you're likely to find in a forensic laboratory.
Biochemistry, also known as physiological chemistry, is the study of the chemical structures and reactions in biological systems -- living things. Biochemists investigate such things as the chemical processes of diseases and those of digestion. As such, biochemists often do research that is related to medicine.
As knowledge of chemical processes develops, these traditional subdisciplines of chemistry become yet more subdivided. For example, a physical chemist might choose to focus on the physics of biological reactions in a field known as biophysical chemistry; an organic chemist could also apply the concepts traditionally reserved for physical chemistry in what is known as physical organic chemistry; or an analytical chemist might specialize in analyzing substances in the environment in the field of environmental chemistry.