A swashplate is a device that translates the pilot's commands from the non-rotating fuselage to the rotating rotor hub and mainblades. It is the lowest part of the rotor head.
The swashplate consists of two main parts: a stationary outer ring and a inner ring that rotates with the main rotor mast. These rings are connected by a bearing. There is also a ball joint inside the rotating (inner) ring that allows the whole swash plate assembley to tip in any direction on the main shaft. The stationary (outer) ring is connected to pushrods that lead down to the servos. The rotating (inner) ring has pushrods that connect upwards (usually via some mixing arms) to the blade grips.
The outer swash typically has an anti-rotation pin which runs in a slider or slot to prevent it from rotating, while a phase link drives the inner inner at the same speed as the blades.
In the above photo, the anti-rotation pin has been replaced with pushrods that prevent the outer ring from rotating.
Cyclic blade control
Cyclic controls are used to change a helicopter's roll and pitch. Push rods or hydraulic actuators tilt the outer swash in response to the pilot's commands. The swashplate moves in the intuitively expected direction - tilting forwards to tilt the rotor 'disc' forwards, for instance; but 'pitch links' on the blades transmit the pitch information ahead of the blade's actual position, giving the blades time to 'fly up' or 'fly down' to reach the desired position. This creates a difference of lift around the blades, and the helicopter will tilt towards the side with lower lift.
Collective blade control
To control the collective pitch of the main rotor blades, the entire swashplate can be moved up or down along its axis as well as changing angle for cyclic control. This is called cyclic/collective pitch mixing or CCPM. This mixing can either be performed electronically or mechanically.
Not all helicopters use CCPM, some have a seperate servo for collective pitch, usually driving a wire up the center of the main shaft.