Since we do business all over the world, we get this question a lot. In fact, we're amazed at how many people in North Carolina don't know the following story!
During the early days of this country, "naval stores" were one of the principal industries of North Carolina. Because we're abundantly blessed with native pine forests, it was only natural that many colonists tapped the trees for turpentine- used to make tar and pitch, for the sealing of ships' hulls and a host of other maritime applications (my own ancestors worked in the turpentine and timber industries in those days).
During a key battle in the War Between the States (that's what we call it here in the South), General A.P. Hill's Confederates were making a particularly valiant stand in the face of a vicious Union charge. An officer who was observing the tenacity of one of the North Carolina regiments remarked. "Look at those North Carolinians- they look like they've got some of that TAR stuck to their heels!"
From then on, they were called the Tarheels, and the nickname is now applied to every true North Carolinian. We're called the Tarheel State!