Not necessarily. The state is an organization with a monopoly on the use of legitimate violence that stands above society and gives orders to the people living within it. It uses various bodies of armed people and coercive institutions (courts, prisons, etc.) to force it's subjects to obey it's orders. The literal definition of "democracy" is "the rule of people." There is no reason why "the rule of the people" MUST have a state, although most versions of "democracy" proposed involve a state. The purest form of democracy possible would have no hierarchy whatsoever and would thus not have a state.
Joe R. Golowka JoeG at ieee.org Anarchist FAQ - http://www.anarchistfaq.org
"The State is a condition, a certain relationship between human beings, a mode of human behavior; we destroy it by contracting other relationships, by behaving differently" - Gustav Landauer