First things first, someone has to order the DNA test kit. Each test kit has a unique kit number. For someone who insists on absolute anonymity, someone else, a cousin or friend, can order the kit for the person to be tested. You could order several kits without providing names of those to be tested. We prefer FTDNA because it also offers Y-DNA testing, but all of the major players are reputable. They all usually offer a reducted price several times a year. You should be able to find a price of $49 for a kit close to mothers' day, fathers' day and Christmas, and occasionally even a "summer sale."
The tests themselves are simple and painless. Most involve using a type of swab. You use the swab to collect cells from the inside of your cheek. Just rub the swab against your inner cheek for about half a minute, and you've collected your sample to send to the testing company. (URL: https://people.howstuffworks.com/culture-traditions/genealogy/use-dna-testing-genealogy-research.htm)
Once the sample has been returned to the testing company, the DNA will be analyzed and results returned within six to eight weeks. The results are sent to the e-mail address associated with the original order - meaning that the person being tested can remain anonymous. (Although the testing company requires assurance that the person tested has agreed to be tested and wants to know their name.)
People who are very interested have convinced as many as 40 cousins to be tested, all of whom remain anonymous since the testing company has no contact information for those 40 people. That interested person is the sole point of contact for all of those 40 people.
The next step is to look at your results. The standard complaint is that it is too confusing. People grasp onto the "ethnicity estimates" because these are simplest - - - but not particularly accurate. Among other things, your results will include a list of other customers who have segments of their DNA that match yours. More and bigger matching segments indicate a closer cousin. Fewer and smaller segments most likely means that your ancestor in common is much further back in time.
That interested person who convinced 40 people to test is looking quite far back in time for unknown ancestors - the 1700s. Looking that far back is no walk in the park. It takes a lot of work, a lot of education, and a ton of patience and persistance, but it is the only way to discover these remote ancestors. The good news is that you don't have to do that yourself. Nearly every family has some distant cousin with that knowledge and patience who is already doing that. (My sister, she of the 40 cousins, has discovered 3 "new" ancestors born in the 1750s and disproved a couple others.)
But that knowledgable person needs your DNA to help them with their analysis. Any one person can have hit the DNA inheritance jackpot for one of their ancestors and have gotten a significantly bigger segment of DNA from that one ancestor. And that might just be exactly what the knowledgable person needs to finally break down the "brick wall" for that family, in this case Nimrod and Phoebe Taylor.
"Every DNA segment that we have originated someplace. First, from one of our parents, then from one of our 4 grandparents, and so forth, on up our tree. The further back in time we go, the smaller the segments from those more distant ancestors become, until we have none for a specific ancestor, or at least none over the matching threshold.
"The keyword in that sentence is segment, because we can assign or attribute DNA segments to ancestors. When we find that we match someone else on that same segment inherited from the same parent, assuming the match is identical by descent and not identical by chance, we then know that somehow, we shared a common ancestor. Either an ancestor we’ve already identified, or one that remains a mystery.
"Those segments can and will reveal ancestors and tell us how we are related to our matches.
"That’s the good news. The bad news is that not every vendor provides segment information. For example, 23andMe, FamilyTreeDNA, and MyHeritage all do, but Ancestry does not.
"For Ancestry testers, and people wishing to share segment information with Ancestry testers, all is not lost.
"Everyone can download a copy of their raw DNA data file and upload those files to vendors who accept uploads, including FamilyTreeDNA, MyHeritage, and of course GEDmatch."
(copied from https://dna-explained.com/2021/10/18/autosegment-triangulation-cluster-tool-at-gedmatch/)
And that is the reason that you want to upload your "raw" DNA data file to GEDmatch. You will be joining the ranks of the experts who have tested with any of the several DNA companies.
Now, supposing that you'd like to see how this works, pick your most recent ancestor whose surname is unknown. You are hoping to find someone already tested who descends from that person's extended family.
As an example, suppose that is your grandmother's grandmother (4 generations back), which means you are looking for third or fourth cousins descending from someone born around 1830. Third cousins usually share between 30 and 230 cM of DNA, averaging 73 cM; fourth cousins share an average of 13 cM of DNA, so you are looking for anyone on your list of matches with roughly that much DNA in common - and there are likely nearly a thousand "fourth cousins." Here is a link to a good explanation of 4th cousin matches: https://whoareyoumadeof.com/blog/what-are-4th-cousin-matches-on-ancestry-dna/. (A third cousin would be descended from that grandmother's grandmother; a fourth cousin would be descended from one of her siblings - and that fourth cousin would likely have a different surname - which is what you want to find.)
If you have DNA from a paternal cousin and a maternal cousin, you can often use those to decide to whether your match is on the paternal side or maternal side of your family.
Similarly if you have DNA from a couple of cousins descended from that grandmother's grandmother in question, you can use those to decide if that new match is likely related to her. (Now you can begin to understand why someone would convince 40 cousins to test their DNA.)
If you find someone who seems likely to be descended from a relative of that grandmother's grandmother, you can examine the tree posted by that match (hoping that there is one) to see if you find any names or places that are familiar to your grandmother's grandmother. (And now you can understand the importance of uploading your tree to GEDmatch.)
If you are lucky enough to find a place or surname that is familiar on the match's tree, that is a good place to start looking for information about your grandmother's grandmother. You could also contact the person directly to ask if they have any information that might help you identify your grandmother's grandmother - but don't be disillusioned if they do not reply - the courtesy of a reply is a forgotten art in today's world.
Here is a link to good explanation by Roberta Estes (a great author of DNA information) of how to use one of the new tools offered at GEDmatch:
https://dna-explained.com/2021/10/18/autosegment-triangulation-cluster-tool-at-gedmatch/