Every school child in America has heard the term "Hessian" in the context of Washington having crossed the Delaware to attack them on Christmas night in 1776. Few people know anything about them, or, if they do, they have a rather "cartoonish" image. In some circles the term has become synonymous with "soldier of fortune," perhaps because, even in the 18th Century, they were referred to as "mercenaries".
We must begin by noting that our current concept of a unified, German nation (in no way to be confused with the more recent fall of the Berlin Wall), is a product of the 1870's. The Germany of the 18th Century was what was left of the old "Holy Roman Empire" of the Middle Ages. It was a somewhat bewildering collection of separate, and autonomous, city-states, duchies, and principalities as large and important as Prussia, and as small and nearly forgotten as Anhalt-Zerbst.
When trying to understand their role in the American Revolution, it is important to recognize that one of these principalities was Hanover, which was governed by it's "Elector". That "Elector" happened to be George III of Great Britain (hence the current ruling family of Britain is still, technically known as the "House of Hanover"). Great Britain traditionally relied on its impressive, defensive "moat" — the English Channel, and always maintained a relatively small army in peacetime. These German city-states, being a part of Continental Europe with few natural boundaries, of necessity, had to maintain comparatively large, standing armies for their own safety. Many regarded Great Britain as a natural ally and fought alongside her as such during the Seven Years' War and earlier conflicts.
Being a Constitutional Monarchy, Britain was one of the more liberal governments on Earth. The economy was generally in good shape, so labor was almost always at a premium. Consequently, Britain always found it difficult to raise many new regiments for it's army during times of war. Those new men it did recruit (or impress), needed to be trained. The new regiments needed to be exercised with others to become militarily viable. The German princes would often see Britain's military needs as opportunities to, at the same time:
The soldiers were themselves only mercenaries in the sense that they were paid for their service to their own ruler, just as any member of the U.S. Military is today. A "soldier of fortune" would be someone who acted, on his own, in a sense as a "free agent", contracting his services to whoever would pay him. In all, George III was able to contract with six of his fellow German princes for the use of some of their troops:
All of these regiments were under the command of Col. Johann Gottlieb Rall. As grenadiers and fusiliers, they considered themselves elite units and thus, all the enlisted men wore a form of the well known, tall brass miter cap.
These units had already served with valor at the Battle of Long Island, the Battle of White Plains, and the taking of Ft. Washington in the several months prior to their occupation of Trenton.