Quilting cottons are considered standard cottons. They have a little stretch across the bias (diagonally) and minimal stretch on the grain. This is what most people refer to when they say to use "cotton" fabric. Our most common quilting cottons are poplin and broadcloth.
Flannel is another common quilting cotton. Like poplin and broadcloth, it is a woven cotton, but flannel has nap which makes it a fuzzy and cozy fabric.