From the introduction:

Intro: What Is Biochemistry?

Biochemistry is the study of the chemistry of, and relating to, biological organisms. It forms a bridge between biology and chemistry by studying how complex chemical reactions and chemical structures give rise to life and life's processes. Biochemistry is sometimes viewed as a hybrid branch of organic chemistry which specializes in the chemical processes and chemical transformations that take place inside of living organisms, but the truth is that the study of biochemistry should generally be considered neither fully "biology" nor fully "chemistry" in nature. Biochemistry incorporates everything in size between a molecule and a cell and all the interactions between them.

Biochemistry essentially remains the study of the structure and function of cellular components (such as enzymes and cellular organelles) and the processes carried out both on and by organic macromolecules - especially proteins, but also carbohydrates, lipids, nucleic acids, and other biomolecules. All life forms alive today are generally believed to have descended from a single proto-biotic ancestor, which could explain why all known living things naturally have similar biochemistries. Even when it comes to matters which could appear to be arbitrary - such as the genetic code and meanings of codons, or the "handedness" of various biomolecules - it is irrefutable fact that all marine and terrestrial living things demonstrate certain unchanging patterns throughout every level of organization, from family and phylum to kingdom and clade.

Biochemistry is, most simply put, the chemistry of life.