Cousins - what's it all about?
The dictionary is quite clear on the definition of a first cousin:

One related by descent in a diverging line from a known common ancestor, as from one's grandparent or from one's father's or mother's sister or brother.

I think that most of us are clear on who our first cousins are. In simple terms, they are the son or daughter of an uncle or aunt. In other words, if a cousin (aka first cousin) shares a grandparent with another person, then the two persons are cousins (First Cousins).

All clear so far? Good. Now lets look at the definition of second cousins. The dictionary defines second cousins as....

A child of one's parent's first cousin.

It sounds like this is getting complicated so lets try and simplify it. Second cousins are related through common great-grandparents. So, two people who aren't siblings but who both share the same great-grandparents, are second cousins. The more you think about and apply the concept, the clearer it will become.

Ready for the next step? Third cousins are defined by the dictionary as....

A child of one's parent's second cousin.

We can also simplify this term. Third cousins are related through common great-great-grandparents. In this case, two people who aren't siblings but both share the same great-great-grandparents, are third cousins. And so it goes on; fourth cousins share the same great (x3) grandparents, etc.

Ok, I've got that bit but what does once or twice removed mean? The number of times removed by, is defined by the difference between the number of generations to the common ancestor. This is how the Encyclopedia of Genealogy describes it:

Once removed, twice removed, etc., are used to indicate that the cousins are not the same number of generations from the common ancestor. If the common ancestor is a grandparent of one cousin, but the great-grandparent of the other, then there is a one-generation difference and the two people are 1st cousins once removed.

Blood relationships are determined by computing the shared ancestry of two individuals. Typically, people express such relationships using terms that reflect the most direct relationship. That is determined by counting the generations from each individual to the closest common ancestor; siblings share a parent, first cousins share a grandparent, etc.

Here's how to calculate the relationship:
The closest common ancestor is the Grandparent of the Subject (two generations) who is also the great-grandparent of the Cousin's Child (three generations).

When the number of generations to the common ancestor is not the same for both the subject and the relative, the first part of the relationship term is determined by the lesser of the two distances to the common ancestor. In this case, that is two, and so the first part of the relationship is First Cousin. The second part of the relationship term is determined by the difference between the two distances to the common ancestor. In this case, that is one (three minus two), and so the relationship term is First Cousin, Once Removed.

After reading the above, one might be tempted to call a niece or nephew a sibling once removed, but I don't recommend it! Please note that a descendant of your first cousin can never be a second cousin to you. The number of times removed increases with each generation. See the next tab -- Chart -- to see a visual representation of relationships.

The chart below shows an interesting characteristic of relationship terms. Most of the terms for near relations are not reciprocal: mothers and fathers have daughters and sons, aunts and uncles have nieces and nephews. The most common reciprocal term is cousin: cousins have cousins. Note that 1st Cousin Once Removed appears twice in the chart. The child of your 1st cousin and the child of your great aunt or uncle are both your first cousin once removed. Note that the number of generations difference is the same.

The relationship from a child of your great aunt or great uncle to you is the same as the relationship from you to a child of your 1st cousin. You can use the chart below to calculate the relationship between two people.

Count the generations between the common ancestor and person 1. Use that number to find the proper column. Then count the generations between the common ancestor and person 2. Use that number to find the proper row. The relationship is shown at the intersection of the column and row.