A double first cousin is a person who shares both sets of grandparents with another person. This means that they have the same grandparents, but not the same parents. Double first cousins are more closely related than regular first cousins, but not as closely related as siblings.
So how is this relevant to genealogy research? Well, if you’re trying to trace your family history, knowing that you have a double first cousin is vital. Genetic genealogy, or family history using DNA, isn't an exact science. It seems like it will give you concrete answers but all that it conclusively shows is you share DNA with someone and therefore have a shared ancestor somewhere in your past.
The point of genetic genealogy is to find your ancestors. You do this by finding common ancestors with your DNA matches and then using that information to find other matches who share the same segment of DNA, but you don't know who the common ancestor is.
Once you know which branch you're working on, through the shared segment, this will help you narrow your options. You can also find matches with that same segment of DNA who have a common ancestor with each other, but not you, and then you see if you can connect the ancestors of the common ancestor to your tree. Basically, you narrow your options down through finding common ancestors.
If you have a double first cousin, or any relative with more than one (unexpected) common ancestral couple, this causes a problem with the expected way to do genetic genealogy.
You expect to have one shared common ancestral couple with your first cousin. But a double first cousin, you have two common couples. Normally that situation is a sibling, your parents are what is referred to as the common ancestral couple instead of the grandparents. But with double first cousins, you don't share parents but an extra common ancestral couple.
A double first cousin is important to identify because you will have more shared DNA with them and you will match on both your father and mother's side. If you think this is a regular first cousin, this will cause lots of confusion.
However, a double first cousin is so close, once you identify them, they should be an asset to your genetic genealogy. You can work with their DNA results like they were your sibling although they may have DNA you and your parents didn't inherit.
When you have more distant types of double cousins, it can be far more difficult. This happens when people from different branches of your tree had children. This can be a cousin marriage in your family tree, in the family tree of a match, or just intermarriage in a small community. This situation is extremely common in the U.S. south and in small communities.
These "small communities" can be a small geographic area, especially islands where it is more difficult for people to come and go, but it can also be a community of people that choose to interact with a limit number of the local population. Examples of groups that choose to interact with a limited population are religious groups or cultural groups (such as immigrants from a specific country or region). Clearly these situations are not uncommon!
The standard process for doing genetic genealogy---where we use common ancestors to help find our own unknown common ancestor---is designed for situations where you do NOT have double cousins. If you follow the normal process and share a paternal ancestor and a maternal ancestor with a match and don't realize it, you can be using your shared matches and looking in the wrong branch of your tree for the common ancestor. Often double cousins aren't on the maternal and paternal side, which we are more likely to notice when looking at our matches. They are often in two adjoining branches farther back in our tree.
It is also possible, and sometimes likely, to share more than two common ancestors with a match or group of matches. In the most extreme case, this is called endogamy.
It's important to realize it is possible and sometimes even likely to share more than one common ancestor with a match. It's also important to know that you don't inherit DNA from every ancestor. Some of the unexpected additional common ancestors will be so far back in your tree you don't have DNA from them. This doesn't make the situation any easier, though. You might identify two, three, four, or more common ancestors but only find you share one segment of shared DNA with a match. This doesn't let you know which of those common ancestors the segment comes from, and it might even come from a common ancestor you haven't identified.
Genetic genealogy requires we consider all the possiblities, which includes understanding we could have double cousins. Once you're aware these situations could exist, you can look out for them. You'll be able to carefully test various hypotheses and use your DNA results to find new ancestors.