"With the nap" means that when you run your hand along the fabric it is soft. "Against the nap" means that the fabric is a little rougher when you run your hand. To "brush the nap" I usually go "with the nap".
Velvet dresses and microfiber couches are a great example of fabrics with nap. When you run your had in one direction the fabric looks lighter, run your hand the other way the fabric is darker.