How can a helicopter be designed without a tail rotor?
The torque in a helicopter is generated by the engine driving the main rotor in one direction, which causes the fuselage to spin in the other direction. The tail rotor shaft is horizontally mounted and hence creates it own 'lift' to provide anti-torque. The pitch of the tail rotor gives directional control.

Common ways to design a helicopter without a tail rotor:

Tandem rotors : Two main rotors mounted one in front of the other. The rotor discs spin in opposite directions. All power from the engines is used for lift. Example: CH-47





Transverse rotors : Two main rotors mounted side-by-side. Example: Kamov Ka-22





Co-axial rotors : Two rotor discs are mounted one on top of the other. The engine units power both rotors. This structure requires a very complex swashplate mechanism. Example: Kamov Ka-25





Intermeshing rotors :  In this main rotor assembly, the two set rotor masts are installed at a slight angle to each other, in a transversely symmetric manner, so that the rotor blades intermesh without colliding with each other. Example: Kaman K-MAX





Tip jet : This is an interesting solution to the no-tail-rotor problem. Instead of driving the rotors with a turboshaft engine, compressed air is sent through nozzles installed at the tips of rotor blades. The engines effectively push against the air rather than the helicopter fuselage. The rotor spins just like a Catherine wheel. Example: Hiller Hornet





NOTAR : NOTAR - NO TAil Rotor uses a fan inside the boom to build a high volume of low-pressure air, which exits through two slots and creates a boundary layer flow of air along the tailboom utilizing the Coandă effect. The boundary layer changes the direction of airflow around the tailboom, creating thrust opposite the motion imparted to the fuselage by the torque effect of the main rotor. Example: MD-900





Refer to Wikipedia for more!