This idiom or expression means:

Stingy.
One who doesn’t spend money easily.

Q. Should we ask your brother to help us with the gas money?
A. No, he’s such a penny pincher. He won't give us anything.

Also:
Tight pocket.
Tight with money.

This idiom or expression is short for “I’ll give you a penny for your thoughts,” which means: What’s on your mind? What are you thinking about?