Biochemistry, sometimes called biological chemistry, is the study of chemical processes within and relating to living organisms. By controlling information flow through biochemical signalling and the flow of chemical energy through metabolism, biochemical processes give rise to the complexity of life.
Today, the main focus of pure biochemistry is in understanding how biological molecules give rise to the processes that occur within living cells, which in turn relates greatly to the study and understanding of whole organisms.
Depending on the exact definition of the terms used, molecular biology can be thought of as a branch of biochemistry, or biochemistry as a tool with which to investigate and study molecular biology.
Much of biochemistry deals with the structures, functions and interactions of biological macromolecules, such as proteins, nucleic acids, carbohydrates and lipids, which provide the structure of cells and perform many of the functions associated with life.
The chemistry of the cell also depends on the reactions of smaller molecules and ions. These can be inorganic, for example water and metal ions, or organic, for example the amino acids which are used to synthesize proteins.
The mechanisms by which cells harness energy from their environment via chemical reactions are known as metabolism.
The findings of biochemistry are applied primarily in medicine, nutrition, and agriculture.
In medicine, biochemists investigate the causes and cures of disease. In nutrition, they study how to maintain health and study the effects of nutritional deficiencies. In agriculture, biochemists investigate soil and fertilizers, and try to discover ways to improve crop cultivation, crop storage and pest control.
source: adapted from http://en.wikipedia.org/w/index.php?title=Biochemistry&oldid=603469180
Commentary, a link between biochemistry and molecular biology
Traditional biochemistry tends to be about the chemistry of living cells. This essentially means using the techniques and principles of chemistry and applying in to what a cell does. For example it deals with things like concentrations of chemicals within parts of the cell. In particular things like the concentrations of ions like sodium Na+ H+ which lends it self to the concept of pH levels in the cells. It also deals with chemical reactions within cells. How one molecule ends up as another. It is less concerned with how these things are facilitated but more about measuring changes of chemicals within the cell.
It is molecular biology that really starts to investigate the causative mechanisms of the chemistry. For example the changes in pH in a mitochondria are the result of a protein in the membrane called a proton pump. The biochemist will say there is a pH difference within a mitochondria and there may be some chemicals that enhance/disrupt this process. The molecular biologist will look at the membrane and investigate what is going on there and link it back to what the biochemist has observed.