The mesh is where the quarterback puts the ball into another players hands or against his stomach as part of a handoff or a fake handoff.
The quarterback’s job is to put the ball on the running back’s stomach. The running back is not supposed to reach for the ball. He puts his forearms parallel to the ground with his arm closest to the quarterback high and his other arm down low. The quarterback should be able to slide the ball in (and out) of the pocket he creates. The running back should roll the top arm and the top of his body over the ball to take the handoff.
In the case of a fake, either because of an option or play action the running back still rolls over the ball, but the quarterback slides the ball back out.
Gridiron Glossary is a resource for football terms that are often used by commentators, coaches and players but rarely defined. If there is a term you have questions about or a definition you don’t agree with, let me know at firstname.lastname@example.org.