I am Uncle to my niece. What relation is my niece to my cousin?

Depends what kind of cousin you're talking about. Let's go with a first cousin. That is, you and your first cousin share one pair of grandparents.

Your niece is your cousin's first cousin, once removed.

Adam and Beth have
a daughter, Carol, and a son, David.
Carol's child is you, Ed. David's daughter is Fran. Fran is your first cousin.

Let's say Carol has a second child, George. George is your brother.
George's daughter, Heidi, is your niece.

If Fran has a child, Iris, then Heidi and Iris will be second cousins.

http://lifehacker.com/second-cousins-once-removed-and-more-explained-in-1661572056


According to the chart, it looks like your niece would a first cousin once removed.

This is an interesting question, with different answers depending on your cultural background.

Yeah, we refer to my mom's first cousins as aunt and uncles.

I guess 'cause they're older, it's more respectful to call them by a title.

Also, saying "Aunt Olivia" is less cumbersome than saying "First Cousin Once Removed Olivia."

But then again, it's a Hispanic convention.

Tia Olivia.

It's the same in German speaking countries. (And very similar in all countries following (somehow) the Roman law.)
But then again, it differs in the (colloquially) spoken language from region to region ...

Ma cousine germaine lives in France, but she's actually German.