The answer in a nutshell is microevolution combined with the fact that DNA can mutate.
Longer answer: The enzymes that make new DNA for each cell in the body sometimes screw up, leading to mutations. A mutation is just a changed DNA sequence. Mutations often do nothing. Sometimes, though, they can change the attributes of offspring. (The mutations only matter evolutionarily when they occur in the sperm or egg, because they control the childrens' entire genetic code.) If the mutation has a negative impact, the mutant offspring may not live long enough to reproduce, thereby preventing the mutation from being passed on to another generation. If the mutation is good, though, the offspring will live longer and/or reproduce more, causing the new trait to be passed on to many children, ensuring that the future generation have this new mutation. After many generations, the resulting organisms have plenty of mutations making them different from the original parents (Adam and Eve if Genesis is literally accurate).
By the way, when I talk about mutations, I'm referring mainly to reproductive cells. Don't worry that you will mutate during your life and become someone else. Mutations occur in individual cells, but all the cells in your body have nearly identical genetic sequences. (I'm talking about something like 99.999999% similar.) However, when a cell has a mutation in a section of the DNA known as a protooncogene, the rrsult is cancer.
There is also the possibility that there were more than 2 original people. (This scenario would not rule out the Genesis version because God may have created others. Keep in mind that Adam and Eve had no daughters mentioned in the Bible, so either Eve was guilty of incest, which God would surely be displeased with, or there were some other people that were not included in the Bible.) If many people existed at first, that would allow for more variation, but microevolution would still cause many changes from the original generations. Furthermore, geographic isolation has led to individualized DNA. (After all, it's hard to get Mongolian DNA if everybody in your family lived in England, for instance.)