The original study of evolution by Charles Darwin, which was based on his research on various species in the Galapagos and other islands of the Pacific Ocean, was the first time that evolution was systematically studied among various species. After the work of Gregor Mendel identifying genes as the unit of inheritance, and Thomas Hunt Morgan localizing genetic information to DNA, it became clear that DNA is the driving force of evolution. Mutations in DNA, particularly genetic mutations, provide the variation necessary for natural selection. Mutations create pools of variation in DNA, and the environment and other selective pressures select for specific mutants, which is how evolution occurs.
Most organisms with DNA genomes, such as all eukaryotes, prokaryotes, archaea, and some viruses, have a much lower mutation rate in their genomes than RNA viruses do in their genomes. Most DNA polymerases have mechanisms to fix errors in replication, while few RNA polymerases do. RNA synthesis is often a throwaway process. If you make an RNA that has too many errors, you degrade it or throw it away. DNA takes the approach of actually fixing the errors.
A well-supported theory of evolution is that all life on Earth originated from an RNA-based ancestor, or the "RNA World" hypothesis. RNA is quite amazing because it can both function as a genetic coding molecule and as an enzyme to catalyze reactions. Because of the dual functionality of RNA, many believe that the original "life" on earth was self-replicating RNA molecules. These molecules evolved and began to gain functions, to the point that some organisms began using DNA as their genomes, and had RNAs function as enzymes or coding molecules for proteins.
All of this is speculation, but it is an interesting theory that organisms may require better proofreading, and DNA replication requires more proofreading, which is why DNA became the dominant genetic molecule instead of RNA. Most likely, DNA polymerase developed a low-level mutation rate due to years of selection against lethal mutations. However, the rate of DNA mutation is low enough that there is some level of evolution even within the human population.
One example of human evolution is in the prion protein, which is the causative agent for kuru, a disease of the brain. Kuru is a disease common in Pacific Islanders because it is spread by consumption of brain tissue, and cannibalism is common in that region. Due to the prevalence of kuru in this region, a mutation in the prion protein renders individuals immune to kuru. Therefore, Pacific Islanders have evolved to be even more efficient at cannibalism. Watch out next time you are in Tahiti! Just kidding, Tahitians are lovely—just make sure you have a little extra food on hand.
However, when we think of our genome, so little of it is devoted to actual genes. Why have we evolved to have such a small portion of our genome devoted to genes? One possible theory is that having such a large genome allows us to easily add new genetic information. As complex organisms, eukaryotes more often evolve through duplication of genes and variation of duplicated genes, such as the various immunoglobin genes that make antibodies. Most gene duplication events occur as a function of a retrotransposition event or as an error in homologous recombination. Forming paralagous genes, or genes that have specialized within an organism after being duplicated, might be a more useful adaptation than a single mutation in a given gene, which might explain the abundance of retroelements in our genomes. Viruses and some bacteria, on the other hand, have small genomes, and single mutations in a gene dramatically affect their functions. Altogether, the study of evolution and DNA replication goes hand-in-hand.