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Could you please tell me what these sentences mean, with some examples? Thanks.

1: A first cousin once removed is the child of one's first cousin.
2: The grandchildren of one's first cousin are one's first cousins twice removed.
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Removed refers to generations.
First cousins share the same grandparents.
Second cousins share the same great-grandparents.

Your first cousin once removed could be either your mother's/father's first cousin, or your own first cousin's child.

Let's take the British royal family.
The queen has several children. They are, of course, brothers and sisters. Sister, I guess.
Their children are first cousins to each other. Prince William and Princess Beatrice are first cousins. They share the same grandparents, the queen and her husband.

Now let's assume it's 10 years later and William and Beatrice have each had children.
Their own children share the same GREAT-grandparents (again, the queen and her husband) and so they are second cousins to each other.

William is the "first cousin once removed" to Beatrice's children, as they are to him.
Beatrice has the same relationship with William's children.
Comments  
If you get no better answer, try:

http://www.google.co.uk/#q=family+relationship+chart&hl=en

and choose the one that makes most sense to you!!
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 BarbaraPA's reply was promoted to an answer.
Thanks, everyone.

GG, your explanaion is very helpful. I understand it. But after having a look at M-W I'm confused again. William is first cousin (of a older generation) to Beatrice's children. How can a second cousin's child be your second cousin? Of a younger generation?

removed
1 a : distant in degree of relationship b : of a younger or older generation <a second cousin's child is a second cousin once removed>
2 : separate or remote in space, time, or character
[M-W's Col. Dic.]
I think you misunderstood what I said.

William's child will be called "Little Will" and Beatrice's child will be called "Little Bea."

William and Beatrice -- First cousins
Little Will and Little Bea -- Second cousins

William and Little Bea -- First cousins, once removed. A generation keeps them from being first cousins.
Beatrice and Little Will- First cousins, once removed.

The child of your first cousin is your first cousin, once removed.
Your own child and the child of your first cousin are second cousins.
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GG, thank you very much for the explanation. I understand it now. it seems M-W's definition was confusing.