The most common configuration is the combination of one mainrotor and one tailrotor. The tailrotor will compensate the torque which is produced by the mainrotor. The tailrotor is also responsible for the control of the helicopter along the vertical axis , during hover flight.
The arrangement as a tandemrotor is mainly used with big helicopters. Because of the opposite rotation of the rotors, the torque of each single rotor will be neutralized. The construction of the control system is much more complicated, compared to a helicopter with a tailrotor. The control along the vertical axis during hoverflight is done by bending the rotordiscs against each other.
The arrangement of two rotors side by side was never very popular. This design was used for the biggest helicopter ever built Mil Mi-12, but it was never very successful.
The System with two rotors, comb into one another, like a gearwheel and combines two rotors. Like the tandemrotor, this configuration doesn't need a tailrotor, because the torque is compensated by the opposite rotation of the rotors. This system was developed during the early days of helicopter flying, but fell into oblivion. Today this kind of arrangement of the rotor has been rediscovered and used with the Kaman K-MAX, a single seat helicopter, mainly used for external load transportation.
The last configuration is the coaxial rotor. One rotor is on top of the other and turn in opposite directions. The control along the vertical axe occurs as a result of different lifts of the two rotordiscs. Depending on which rotor produces more lift, the helicopter will turn to the left or right, because of the torque. For these helicopters it is not possible t o reach a high cruising speed, because the drag is too large. Only after the development of the rigid rotor, has it been possible to build the two rotors closer together and reduce the drag considerably. This configuration has been mainly choosen by Russian Kamov helicopters.
The development of the helicopter hasn't stopped. Nowadays there are new configurations like the NOTAR (NO Tail Rotor) and tilt rotor systems that are being further developed and refined. The X-Wing project, where the rotor stops during flight and then works as a "normal" wing, will probably not be ready to go into production for a long time (if at all).