Your common ancestor with another person is the nearest ancestor you both descend from. It is the anchor of every blood relationship Kinage names, drawn as a gold shape on the diagram.
Couples, not people
Ancestors usually come as couples. Two first cousins have two common ancestors: their shared grandparents. Both are equally "the" ancestor, so Kinage lists them together. If only one is shared, the relatives are half relatives.
Doubling back
You have two parents, four grandparents, eight great-grandparents, and so on, doubling each generation. Count back thirty generations, about 750 years, and the sum reaches over a billion slots, far more people than were alive in medieval Europe. The only way out is that the same people appear in many places on your tree. This is pedigree collapse.
Why it matters in practice
In small or close communities, cousins marrying cousins brings a common ancestor back into the tree twice. Such a couple's children descend from that ancestor along two routes, and the figures change. The true shared DNA is higher than the plain table says. Kinage finds the nearest shared ancestors on your tree, and when more than one route exists it names the closest.
The simple count is still the right place to start. Put two people from your tree into the calculator, and read the DNA guide for how the numbers behave.