It's complicated, but not that complicated. The relationship is calculated by the generational distance, and the distance from your own generational position.
Let's start with the "great/grand" part. Every "Great means you are going back one generation. So our parents are 1 generation removed, our grandparents are 2 generations, and our 5th great grandparents are 7 generations back. I skipped a few, but you get the idea - and I have a handy chart below. The same concept exists with Aunts and uncles, so my Great Grand Aunts and Uncles are4 generations back.
Cousins are the children of aunts and uncles. Their children, or their children's cousins, are always your first cousins. The removed part indicates the generational aspect. So the daughter of my 29th great Grandfather (Máel Coluim mac Cináeda) was Donada, my 28th Great-grand Aunt, and her son was my 1st Cousin, 29 times removed. His name, by the way was Mac Bethad mac Findlaích, although history knows him as Macbeth. He and Duncan (everyone remembers "The Scottish Play", right) shared a common Grandfather, which made them cousins.
It gets complicated because, the removed part is based on the distance from yourself, in the shortest fashion, so a cousin that is 1 generation above or below you, is also 1x removed from you. So the children of your 2nd cousin would be your 2nd cousin 1x removed, and their parents would be 1st cousin 1 x removed. Anyone at the same generational level is simply a cousin, with the prefix (i.e. 6th) indicating the generational distance. Again, you can refer to the chart below, but to make it even easier, every entry in the database is tagged with the correct relationship, so simply look up Macbeth, and you can see I am right 1st cousin 29x removed.
OK, lets move on to consanguinity.
Consanguinity - from Latin consanguinitas, which means blood relatives, is defined as people with common kinship. There is a clear relationship between birth defects and close blood relations, so they have been discouraged, typically by Secular law, throughout history.
Both civil and religious laws prohibit marriage based on the degree of consanguinity. Canadian law prohibits marriage between straight line blood relations (siblings, children, parents, grandparents), but has no other restrictions based on consanguinity. The Catholic Church prohibits marriage between any blood relations with a consanguinity of 4 (first cousins) or less, without special dispensation.
Quebec was a quite insular society, and the Catholic dispensations were given as the colony grew, although the incidents were relatively small. In my database, there are only a handful, about 13, cases where I have identified marriage of first cousins.
The Generational chart below, shows the consanguinity relationship.
DNA, short for deoxyribonucleic acid, is the genetic material that carries all information that makes a person an individual, such as the color of eyes or hair, or how your heart works. It is a complex structure chemical substances that are linked together in a chain. It's complex, and I don't want to reproduce material commonly available, so read more on it here.
It is responsible for allowing people to reproduce, and the fact that DNA is passed from parent to child plays a role in genealogy - some practical, and some more hype than fact. The implication given from adds from genealogy companies seem to claim that you can discover who your ancient ancestors were, and while there is a degree of truth to that, the samples of shared DNA diminishes quickly (see the relationship chart below) and would make any direct comparison impossible.
That said, humans share 99.9% of DNA. The differences are called single nucleotide polymorphisms, or SNPs (pronounced “snip”). By comparing these SNPs against know populations or ancestries, companies can make a determination of where they think you have origins. There is a good article explaining this here. Every company that provides genealogy based DNA has there own set of reference groups and algorithims to determine how you fall into certain reference groups. So if you have your DNA tested at each of the big 3 (23andMe, Ancestery and MyHeritage) you'd likely find some similarities, but a lot of differences. Also, these will change over time, as the companies sample sets get bigger, and as genealogy records and family trees improve.
There are many different types of DNA tests, and all provide different types of information, such as haplogroup, which is a grouping of organisms inherited from a single ancestor. haplogroup do not vary a lot over time, and this is what has led 23AndMe to say that I am related to Louis XIV (proven) and Ötzi the Ice Man (I haven't traced my family tree back that far). Although I am certainly related to Louis XIV, it doesn't imply that I am directly related - he isn't, to the best of my knowledge, one of my great-grandfathers. It simply shows that we have a genetic commonality, or a shared relative. Of course, all people of western European descent are related, to a common ancestor born between 600 and 1000 years ago. Family researchers have know for decades that everyone of European descent is related to Charlemagne - and genetic studies have proven that. Just google it - there are more and more studies on this being published every year. to me, the fun is in showing how we are related, not knowing that we are.
Of course, Charlemagne may indeed be my (great) grandfather. He lived about 38 generations back and could have been my 35th Great-Grandfather, and I would have had over 137 billion of them!
I never really answered the question of what my 25th Cousin, 1x removed was, or is.Well it is a pretty distant cousin, 1 generation removed (or in the same age category as my parents) - related to a common relative about 25 generations back. The tie is complicated, but there are all the tools on this site to find out exactly how we are related, but to make it simpler, it is Elizabeth II, Monarch of the United Kingdom (Queen of Canada, too!).