Your first cousin once removed is either

1. the child of your first cousin or the child of your grand-aunt/uncle
2. Why are they 'removed?' ---- Because they are in a different generation than you, when compared to the most recent ancestor that you have in common!

CAUTIONS 

We Americans tend to call the siblings of our grandparents "great-uncle" and "great-aunt." The correct terms are actually "grand-uncle" and "grand-aunt." Go back another generation and we have our "great-grandparents," "great-grand-uncles," and "great-grand-aunts." (Actually, this correct terminology is easier when you get used to it. Everyone in any given generation is referred to with the same number of "grand" and "greats.") 

English kinship terminology .... contains the following principal terms:

1. mother, father, son, daughter, brother, sister
2. uncle, aunt, nephew, niece
3. cousin (differently elaborated in different English speaking cultures)
4. grandfather, grandmother, grandson, granddaughter
5. granduncle, grandaunt, grandniece, grandnephew (in many dialects)
6. plus great-grandmother, great-great-grandmother etc. and great-grandfather, great-great-grandfather etc.
   there are also the affinal terms:
7. wife, husband, brother-in-law, sister-in-law, mother-in-law and father-in-law as well as uncle and aunt. 

From: Representing Anthropological Knowledge: Calculating Kinship: Analyzing and Understanding Cultural Codes http://www.era.anthropology.ac.uk/Era_Resources/Era/Kinship/kinIntro.html By: Michael D. Fischer

Be aware that the term "cousin" was used loosely during an early period of American history, and could be referring to a cousin, uncle or other relative. Americans tend to call the children of their first cousins, their own "second cousins."

That is wrong. Remember the rule about counting the number of generations to the common ancestor to see if there are any "removeds" to add to the cousin title. In this case, the children of our first cousins are our "first-cousins once-removed."

Hope this helps!

In Other Words In case my explanations still have not made it clear, I am quoting below the clever explanation of Duane F. Alwin. The word "cousin" has a variety of meanings, some of which are more precise than others.

We often use the word in a general way to refer to any collaterally related persons more distant than siblings who share a common ancestor. When we want to be more specific, we use the term in a different way: cousins (or first cousins) are the children of siblings. That is to say, the children of my aunts and uncles are my first cousins. Second cousins, on the other hand, are the children of first cousins, and third cousins are the children of second cousins, and so on. In other words, my second cousins are the children of my parents' first cousins, and my third cousins are the grandchildren of my grandparents' first cousins. The degree of cousinness, thus, simply follows generational lines, given kinship relations defined by a common ancestor.

By contrast, when one crosses generational lines to express relationships among cousins in an adjacent generation or across several generations, one normally expresses these cousin relations as "once removed" or "twice removed" according to how many generations separate the related individuals. Thus, one is a first cousin once removed (1C1R) to his or her parents' first cousins, or to the children of his or her first cousins.

I have always gotten a kick out of telling people that I am a cousin to myself.

My maternal grandparents were first cousins once removed -- my grandfather married the daughter of his first cousin. His cousin was 15 years his senior and he was a few years older than my grandmother. In any event, following the above definitions - second cousins are the children of firstcousins - we can see what may appear to be a contradiction. Because they are both daughters of first cousins, my mother is a second cousin to her own mother.

This makes me a third cousin to my mother, as she and I are both children of second cousins inthe same ancestral lineage. And, of course, to myself I am a third cousin, once removed (3C1R).
Thus, when I use my genealogy software to print out the descendants of Samuel CHACEY (our common ancestor) I appear twice (and in different generations)--once as a descendant of my grandfather and once in my grandmother's line.

What better proof that I am my own cousin. Matings between cousins are called consanguineal, meaning that the members of the pair have one or more common ancestors. In some geographical areas at some times such matings can be quite common.

Whether we know it or not, each of us probably has some consanguineous marriage in their pedigree. Most cultures have rules that regulate the degree of relationship permitted between two individuals who wish to marry.

In many societies, including our own, marriages between first cousins, uncles and nieces, and aunts and nephews, are typically discouraged or in some cases outlawed. Although it would mean fewer grandparents to keep track of, such matings are probably not a good thing. Individuals with rare recessive sex-linked traits are often the offspring of such matings. On the otherhand, such consanguineous matings are not necessarily undesirable.

Charles DARWIN married his first cousin Emma WEDGWOOD, and the entire Darwin-Wedgwood lineage was highly inbred. Some have speculated that the pre-eminence of this lineage in the arts, sciences, and the professions may have resulted from some inbred genetic trait. But this is probably the exception, and genetic diversity in families is probably healthier over the long run.

SOURCE: Written by Duane F. Alwin, dfa@umich.edu. Previously published by Julia M. Case and Myra Vanderpool Gormley, CG, Missing Links, Vol. 5, No. 38, 20 September 2000. RootsWeb: http://www.rootsweb.com/ (Duane F. Alwin is Professor of Sociology at the University ofMichigan, where he teaches social psychology, the family andresearch methods. In his spare time he actively pursues thehistory of his own family.)