Notes I took from Professor Ahern's biochemistry class:
Sulfur, phosphorus, oxygen, nitrogen, carbon, and hydrogen account for the main six elements found in every single living organism we know of.
These elements can make multiple chemical bonds or share electrons to make covalent bonds. This is necessary for making long-chain biomolecules. Other elements abundant in the environment (aluminum, iron, sodium, etc.) cannot do what these elements can do.
Carbon's ability to make four bonds also makes it central to the construction of large and complicated bio-molecules. No other elements in the environment have the ability to from four, complex covalent bonds like carbon. A notable exception is silicon, but carbon is probably a better atom for building larger molecules. Some scientists wonder if silicon-based life is possible on exoplanets.