In genealogy exactly are these relationships?

Here's a great online tool that will graphically display your connection to 2nd, 3rd cousins etc.

http://www.wolframalpha.com/input/?i=3rd+cousin

It will also take questions such as :

"uncles, brothers wifes, brother"

What exactly is a second cousin, third cousin, etc., in relationship to me?

You and the children of your parents' first cousins will be second cousins.

You and the children of your parents' second cousins will be third cousins.

What does "once removed" mean?

One generation apart. It goes both ways. Your parents' first cousins are your first cousins once removed. The children of your first cousins are your first cousins too - once removed. Cousins of grandparents and grandchildren of cousins are twice removed. Your 4th great grandfather's cousins are yours 6 times removed.

What is the relationship of my niece's children to me?

grand niece and grand nephew.

What would my cousin's children be called in relationship to me.

Answered above.

Brother, sister, and sibling all mean person with the same parents.

First cousin means person with the same grandparents.

Second cousin means person with the same great-grandparents.

Third cousin means person with the same great-great-grandparents.

Once removed means "child of". For example, if two relatives are "first cousins once removed", then one of them is the child of a first cousin of the other.

Your niece's children are your grandnephews (if male) or grandnieces (if female). You are their great-uncle (if male) or great-aunt (if male).

Your cousin's children are your cousins once removed. You are their cousins once removed.

"Once removed" refers to cousins in different generations. For example, your mother's first cousin, who is a member of her generation, is your first cousin once removed.

You and the children of your mother's first cousin are second cousins. Your children and the children of your second cousins would be third cousins.

However, you and your second cousin's children would be second cousins once removed.

http://genealogy.about.com/library/nrelationshipch...

This may help you work out who is who.