All humans have 23 pairs of chromosomes, with one of each pair coming from our fathers and one coming from our mothers. The 23rd chromosome pair, the one that determines our gender, is unique in a way that makes it particularly useful to genealogy. The chromosome from the mother will always be an X, but the chromosome from the father may be either an X or Y. If the father contributes an X, the child will be a female and have an X-X pair. If the father contributes a Y the child will be male and have an X-Y pair. Thus the Y-chromosome is passed down only from father to son, generation after generation. Serendipitously, surnames are also passed down from fathers to sons so the Y-chromosome is a sort of genetic signature for surnames.
The Y-chromosome is also unique in that it is the only chromosome that is inherited as a complete unit. All our other chromosomes are mixtures of elements inherited from all the ancestors of our parents. That is why two children are never alike. But, in males, the Y-chromosome is passed from father to son as a mostly complete unit that is largely immune from the mixing of genetic characteristics. That is, the Y-chromosome of a grandfather, father, and son will all be essentially identical.
However, all DNA is subject to mutation. Luckily, certain specific parts of the y-chromosome mutate very rarely, perhaps every 500 or so generations. Genealogical DNA testing focuses on some of these elements, called "markers", that tend to remain the same through many generations. At the present time, there are tests available that produce a very precise description of either 12 specific markers or 25 specific markers.
Focusing on these specific markers means that the tested markers of a male's Y-chromosome will be identical to that of his male ancestor from hundreds of years ago, barring mutations. That means, for example, that two descendants of the same immigrant ancestor 300 years ago should have identical markers, even if they are descendants of different sons of that ancestor. That is the primary value of this test in genealogy. If two males are tested and have identical results, that means they have a common male ancestor. We can't determine which ancestor they have in common, but we can calculate the probability that their common ancestor was within a specific number of generations in the past. If two males have significantly different results, that means they do not share the same ancestor, at least not within the last several hundred or possibly thousands of years.
The difference between the 12 and 25 marker tests is that the more markers, the more accurate the test. If two males match in 12 markers, there's a 50% probability that their common ancestor was within the last 14 generations, and a 90% probability that their common ancestor was within the last 48 generations. If they match in 25 markers, there's a 50% probability that their common ancestor was within the last 7 generations, and a 90% probability that their common ancestor was within the last 23 generations.
Obviously, the test results for one person have no value. The value comes from comparing results to other participants. Participants who can prove descent from various ancestors can be compared both to one another and to others who have hit "brick walls". We would ideally like to have at least two participants from each of the known Ivey/Ivie/Ivy immigrants in order to establish the DNA profile of each of those lines.
DNA testing is not a substitute for traditional genealogical research. Just the opposite. DNA test results are most useful when combined with traditional research. For instance, the test results can be used to confirm a suspected connection between two lines or disprove a connection. Although it is impossible to pinpoint a common ancestor from the test results alone, you may be able to do so by combining the results with conventional research.
Normally, we want to test the Y-chromosome because it follows the same line as surnames do. A female Ivey/Ivie/Ivy descendant can participate by having a brother, father, uncle, or male cousin take the test. If you can prove a common ancestor with a male Ivey/Ivie/Ivy, have them take the test.
There is a different test, applicable to both males and females, which looks at mitochondrial DNA (mtDNA). Mitochondrial DNA is passed from mothers to their children. Although sons receive the mtDNA, they cannot pass it on to their own children. mtDNA is therefore passed on through a female line of descent, generation after generation. This test can be used to prove or disprove the existence of a common female ancestor. This has limited genealogical value, for obvious reasons.