Reading a family tree is often confusing and on top of it, reading one which relates people to you saying first cousin once removed and second cousin twice removed, is literally head spinning. Here, let me try to explain to you, this concept, which I took approximately 2 days to come around and understand. Before starting to calculate who is who, I suggest you draw a small family tree, not going up nor down, by more than 2 generations. Once you get the concept, you can go ahead with the calculations of a bigger one. Or else, you are sure to be lost in your own tree and may even end up calling your nephew, as your first cousin at a family reunion party.
Definition of First Cousin Once Removed
Technically, first cousin once removed is defined as, "a person who is not related to you directly but who shares a common ancestor at some level of the family history". That is, your nephew or niece is your first cousin once removed or second cousin once removed depending on the relation you share with their parents. Similarly, your grandmother's brother's son is your first cousin twice removed. Confused? Let me explain the formula that works behind this genealogy system.
The genealogy system I am talking about, relates every one to you, in terms of cousins. So, you can have cousins going back to ancient times. What this system does is, it uses degrees to explain the exact hierarchy of the family relationship. The degrees used, define the several generations that separate the cousins. One thing that you have to do while reading is, assume everyone as your cousin. Forget the concept of nephew, grandfather's brother, etc., or you will end up banging your head against the wall. And do remember, that this method applies to only indirect relations. All the direct relations remain the same.
The degree in first cousin once removed, that is the first part of the phrase, describes the least number of generations between the relative and the nearest common ancestor and the once removed/twice removed part, that is, the second part, defines the number of generations separating the person from you. This methodology is slightly different for the people coming above you, than the people below you in the family tree. You will know the difference as you read on.
Method for People Above You
Think of a person with whom you share your grandmom, but not your parent, then that person is your "first cousin". Because that person is to be the son/daughter of your parent's siblings. Now imagine a person with whom you share your great grandmother, means a person who is the son/daughter of your grandmother's siblings (this person happens to be your parent's cousin), that person is your first cousin once removed. Now let me explain how. The number of generations between that person and the nearest common ancestor, is 'one' and this person happens to be grandmom's sibling (your great aunt/uncle). And that common ancestor is your great grandmom. And thus the term - first cousin. Now coming to the "once removed". The removed term is defined by the number of generations that person is above or below you, and that in this case is, one. That person belongs to your parent's generation, one generation above. So, we can safely say that this person is your 'first cousin once removed'.
On the same lines, what do you understand by second cousin once removed? For the first part, a second cousin shares the same great forebears, while the second half (which deals with the gap in generations) is a conflict of one generation.
Now let's explicate third cousin once removed. For the first part, a third cousin shares the same great-great forebears as yours. Coming to the second bit, once removed speaks about generational divergences between cousins, which in this case is 'one'. Thus to wind it up, third cousins once removed indicates a person who has a common great-great grandparent and is separated by one generation; in elementary terms, belonging to your parent's generation.
If you take careful notice, then cousins belonging to a similar generation are not tagged with a 'removed' annotation.
Method for People Below You
Let me try to explain the other rule with an example. Try thinking who will be your second cousin twice removed? Any guesses? It will be your second cousin's grandchild. I know it is confusing, but unwind your mind and read the following sentences. Let me explain how I calculated this relation. It's simple, as you calculate the relations of the generations coming after first cousin, second cousin or third cousin (these will be the people younger to you), you have to follow the simplest rule. Going down, the degree has to remain the same, only the generation gap needs to be changed. So, this relative is your second cousin's grandson and hence, the degree remains second cousin only, but the generation gap is that of two between the person and you and hence, 'twice removed'.
I hope this read gives you some idea on how to find your cousins going back to the generations before you. If not, go grab a cup of coffee and read it once more, along with a handy family tree chart. You will get it eventually. You will come across many cousins (like 15th and 16th cousins 4 times removed etc.) on an expansive scale as your family tree develops. So don't fret your brains as much as I did, there are many software programs made available that automatically name the relatives, so that you can have your data in a jiffy.