In short, the definition of cousins is two people who share a common ancestor. Here are a few definitions of cousin relationships:

First Cousin: Your first cousins are the people in your family who have at least one of the same grandparents as you. In other words, they are the children of your aunts and uncles.

Second Cousin: Your second cousins are the people in your family who share the same great-grandparent with you.

Third, Fourth, and Fifth Cousins: Your third cousins share at least one great-great-grandparent, fourth cousins share a great-great-great-grandparent, and so on.

Removed: When the word "removed" is used to describe a relationship, it indicates that the two people are from different generations. "Once removed" indicates a difference of one generation, "twice removed" indicates a difference of two generations, and so forth.

For example, the child of your first cousin is your first cousin, once removed. That is, your cousin's child would be "almost" your first cousin, except that he or she is one generation removed from that relationship. Likewise, the grandchild of your first cousin is your first cousin, twice removed (two generations removed from being a first cousin).

Many people confuse the term "first cousin, once removed" with "second cousin." The two are not the same.

Keep in mind that you and a relative only need to share one grandparent to be first cousins, or share one great-grandparent to be second cousins, etc. If the ancestor in question had more than one spouse and the two of you are descended from different spouses, you are full cousins. There is no such thing as a "half cousin" although you will hear people use that term occasionally.

The following consanguinity chart may help to explain the relationships:

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Cousins Table

A cousin is someone who shares a common ancestor with you. Use this chart to determine your relationship.

For instance, we will assume that you and your newly-found relative are both descended from John Smith. This common ancestor is your great-grandfather and also is the great-great-grandfather of your newly-discovered cousin.

In the above chart, go across the top to find your ancestor: great-grandfather.
Next, go down the left column to find your cousin's relationship to the same person: great-great-grandfather.

Now notice where the two intersect in the above chart: you and your new cousin are actually second cousins, once removed.

You may prefer to use an automated online tool to determine relationships. Ancestor Search has one that I found simple to use.

Take a look at http://www.searchforancestors.com/utility/cousincalculator.html

Here are a few other terms you may encounter when determining relationships:

HALF - Means you share only one parent. Example: half-brothers may have the same father but different mothers, etc.

STEP - Not blood kin, but a close legal relationship due to re-marriage of a parent, such as step-mother, step-brother, step-son, etc.

DOUBLE FIRST COUSINS - Are first cousins twice, once on your father's side and once on your mother's side, since your father's sibling married your mother's sibling.

IN-LAW - They are not really blood kin but are treated as such because they married blood kin.

Example: Your mother-in-law is not really your mother but is treated as such because you married her daughter/son. In law, you and your spouse are considered "one". Also your brother-in-law is your brother because your parents are also his parents, in "law" (mother-in-law, father-in-law, etc.).

KITH and KIN - "Kith" are friends and acquaintances whereas "Kin" are blood relatives or someone treated as such, in law.

By the way, it is estimated that everyone has approximately 4 trillion 20th cousins! In other words, everyone is related to nearly everyone else.