Easy. Go laterally across at the generation of the elder relative. 1st cousins share grandparents, 2nd cousins share great-grandparents, etc.
Every generation you have to go down to get to the other relative is one more "removed".
Okay, here's an actual easy way to understand it.
Cousins (first cousins) have parents who are siblings. Or to put an easy way, first cousins share grandparents.
Second cousins share great-grandparents.
Third cousins share great-great-grandparents, and so on.
So, if you have a co-worker who has a famous great-great-granddad, and hey, you have the same famous great-great-granddad, then you're third cousins with that co-worker. The once-removed? Those are moves up and down the generational ladder. If the co-worker was only the great-grandson of that famous person to your great-great-granddaughter status, then you'de be once removed from each other.



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Note added at 10 mins (2007-08-11 23:41:45 GMT)
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How to remember all this:
1st, 2nd, 3rd cousin etc. should all be on your level of the family tree, from the same generation you are.
You can be removed in two ways:
1) descendants of your cousins are removed by the number of generations from that cousin, and keep the same 1st, 2nd designation as the cousin they are descended from. Your 2nd cousin's grandchild is your 2nd cousin twice removed.
2) cousins of your ancestors keep the same cousin number they had with your ancestor, but are removed by the number of generations you are descended from that ancestor. Your grandmother's 3rd cousin is your 3rd cousin twice removed.
As a result, the cousin designation is mutual: you are the 3rd cousin-twice removed of your 3rd cousin-twice removed
It's simpler in Spanish.
My father's brother is my uncle. His son is my cousin.
My father's cousin is my "tio segundo" (2nd uncle). His son is my "primo segundo" (2nd cousin).
My father's "primo segundo" is my "tio tercero" (3rd uncle). His son is my "primo tercero" (3rd cousin).
You get the idea