Biochemical homologies provide some of the strongest evidence for evolution — partly because of the level of detail they provide and partly because the nature of some of the homologies makes any explanation other than evolution seem even more farfetched than with the larger-scale homologies. There are a variety of different avenues of biochemical evidence for evolution, but most of them are either examinations of genetics or of proteins — genetic homologies and protein homologies.
The shared biochemical mechanisms are among the widest shared homologies in life. Most of the anatomical homologies described elsewhere are shared by some subset of living organisms, but when we get down to the molecular level we find homologies that cut across all living organisms. If life forms arose independently, whether through natural process or divine intervention, there is no reason to think they would all share these characteristics, especially given that alternatives are known to exist. These biochemical homologies argue strongly for common ancestry to all life.
In a sense protein homologies are reflective of genetic homologies because genes are made from proteins. However, they deserve independent attention because there has been a lot of work done on examining proteins. The understanding of genetics is a relatively newer science, and while more and more data is becoming available regularly, at this point in time proteins may have been studied more than genes.
A protein is a string of amino acids. Proteins range in size from around 50 amino acids to thousands. Proteins are among the most important chemicals in life: in addition to making up a good chunk of the structure of many organisms, proteins are involved in regulating or controlling many of the functions of a living organism. The characteristics of a protein are determined by the sequence of amino acids of which it is constructed.
There is a homology between all livings things regarding amino acids because the same twenty amino acids are found in most kinds of living things. These twenty are a small subset of the amino acids that occur naturally (~250) and there is no known reason why these particular twenty amino acids need to be used over some other subset of amino acids if different life forms had originated independently. However, it does make sense if all life evolved from a common ancestor that happened to use these twenty amino acids.
Homologies can be found not only among the constituents of proteins but also among proteins themselves. It is important to understand that, in many cases, smalls changes in some of the amino acids that make up a protein do not appear to have much if any effect on the functioning of the protein. Thus we can have a set of proteins that do essentially the same thing but are not identical. For some proteins it is estimated that even significant changes in the amino acids from which it is constructed will not affect its function.
Hemoglobin is an example because there are actually several types, all of which serve the function of binding oxygen in the blood and yet differ in their amino acid sequences. Hemoglobin is found in a wide variety of life forms and they are all very similar in structure. Given that there are so many different sequences of amino acids that could make a functional hemoglobin molecule, we can ask why the various types of hemoglobin among vastly different creatures are so similar. Evolution provides a meaningful answer.
Genetic homologies are similarities between the genetic codes of different living organisms which cannot be explained simply by functionality. There are an abundance of particular examples, but the biggest example is the common genetic mechanism itself: Why should all living creatures share the same basic genetic structure?
The presence of the same basic genetic structure in all living things is not chemically determined. It has been shown scientifically that there are (probably many) other alternatives that would work for coding proteins (which is essentially what DNA/genes do). For instance, DNA uses only four bases out of the many naturally occurring bases (>80). It has been shown that other base codings are possible. Why does all of life use DNA with the same four bases?
Why should every single life form on the planet share the same basic genetic mechanisms? It actually goes beyond just the genetics since there are other biochemical aspects which are shared by all life: for example, the common use of ATP as an energy storing molecule. There are other alternatives to this molecule, some of which are even more efficient at storing energy.
The primary creationist argument against biochemical homologies is some sort of "divine" plan, but there is no coherent, scientific argument that would explain this evidence other than the theory of common descent. Non-scientific explanations are, of course, possible and creationist arguments generally fall into this category. However, they do nothing to detract from the evidential support for evolution since evolution operates within the realm of science.