Helicopters depend on rotors for lift, and although rotors are airfoils, they act differently than airplane wings in several important ways. These differences account for several of the major problems early designers had.
The first sucessful rotor systems were designed by Juan de Cierva for his gyroplane. This craft was basically an airplane with the wings removed and a rotor substituted as the lift producing component.
This early rotor system was not designed to be controlled by the pilot the way a modern gyroplane or helicopter is. It simply produced lift, and aircraft control was by the normal rudder and elevator controls. The rotor was unpowered, and simply spun like a pinwheel due to the relative wind blowing through the rotor system. This is called autorotation.
The problem that Juan encountered was that as his gyroplane would start to fly, it would invariable roll to the left and crash. Models he built using lightweight rotors did not experience this. He figured out that the cause was what we now call dissymetry of lift and that he could solve the problem by installing a flapping hinge.
While the flapping hinge solved one problem, it caused another. Once the flapping hinge was installed, rotor blades were breaking due to high stress. The stress was identified as being caused by coriolis force and the installation of another hinge, the lead-lag hinge solved that problem.
With the installation of a hinge or bearing to allow feathering the rotor system would have all the capabilities of a modern rotor system.
Trying to turn the rotor system with an engine causes one problem: torque reaction. In this case, if you try to turn the rotor system with the engine, the equal and opposite reaction called for by Sir Issac Newton will cause the entire helicopter to rotate in the opposite direction. Several methods have been used to counter this effect, the two main ones being either the use of two or more counter rotating rotors, or the use of a tail rotor to counter the rotational force. The other significant methods are tip driven rotors, and the MDHC NOTAR system.
If you've ever played with a gyroscope you've probably experienced gyroscopic precession. A result of gyroscopic precession is that maximum deflection of the gyroscope occurs approximately 90 degrees later in the rotation from where the force is applied. This effects helicopters in several ways, but probably the most noticable is in the rigging of the pitch links and swashplate. Pitch information coming from the pilot's cyclic is actually transmitted to the blade 90 degrees early, so that the maximum tilt of the rotor disk will occur in the direction the pilot has moved his cyclic control. Check out the (missing) section on pilot controls and swashplates for more information.