UPDATED (Sept. 30) - Wikipedia link to article on Cousins that includes a handy reference matrix, reproduced below. Thanks to nrmrvrk for the tip.
[NOTE: I am not the original author of the text below. My cousin sent it to me unsourced, so if the original creator is out there and reading, give me a shout out and I'll give you a credit.]
A term often found in genealogy is "removed," specifically when referring to family relationships. Indeed, almost everyone has heard of a "second cousin once removed," but many people cannot explain that relationship. Of course, a person might be more than once removed, as in third cousin, four times removed.
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.
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.