This can get pretty lengthy, even if we try to keep it simple, but here goes!
Rotating Wings. In a conventional fixed-wing aircraft, lift is generated by moving the wings, with their airfoil section, through the air at a fast enough speed to produce enough lift to fly the aircraft. Aifoils + airpseed = lift (plus drag as well), but we'll ignore that. Now there is no reason why a wing has to move in a straight line to generate lift - we can produce useful lift by rotating the wing. The rotor blades are the "wings" of a gyroplane. The rotate at between 300 and 450 rpm, depending on disc loading and a number of other factors, and they generate lift as they move through the air.
Why they rotate. In flight, the gyroplane rotor is not powered by a link to the engine. The power to operate the rotor comes from the relative wind flowing up and through the rotor disc. Given the angle of the blades to the relative wind and their airfoil section, some of the energy generated as lift is applied to keeping the blades rotating. This is known as autorotation. Although the direct energy source for keeping the blades rotating is the relative wind, the engine is the ultimate energy source since it pushes or pulls the gyro through the air, creating the relative wind. For this reason, a gyroplane must take off like a fixed-wing, running into the wind to build up blade speed to the point where the aircraft will fly. The required takeoff run can be reduced by a prerotator - a temporary link to the engine or other source of power to get the blades started and bring them up close to flight speed.
Unequal Lift. Because the gyroplane is moving through the air, the advancing blade generates more lift than the retreating blade. If the rotor were rigidly mounted, this would cause the aircraft to tip over. In most gyroplanes, the rotor is hinged on a teeter bolt, much like a see-saw. The advancing blade will rise and the retreating blade will fall, automatically balancing the lift generated by the two blades and eliminating any tendency for the aircraft to roll due to unequal lift.
How the Controls Work. The control stick alters the fore and aft and side-to-side angle of the rotor disc. Pushing the stick forward will tilt the rotor disc forward, reducing drag and increasing airspeed. Pull the stick back and the angle increases, producing more drag and decreasing airspeed. Thus, fore and aft stick movements control airspeed, just like a fixed-wing. Side to side movements of the stick tilt the rotor disc from side-to-side, causing the aircraft to roll in response to the sideways movement of the stick - just like a fixed wing. Yaw control is produces with a standard vertical fin with a moveable rudder controlled by pedals - again, like a fixed wing. Finally, whether you climb, hold altitude, or descend at any airspeed is a function of the engine throttle setting.
While a gyroplane is a basic "stick and rudder" flying machine, how you use the controls in specific situations is sometime critically different than how you would react in a fixed-wing. That's the reason why dual flight instruction is essential!