Helicopter flight is actually very simple when broken down into its parts. I'll begin the explanation by going over the various parts of the helicopter so you'll be able to follow me.
Pilot Side Controls
Helicopters are controlled with three different controls:
Cyclic- this is the "joystick" you see in the cockpit. It controls the forward (fore), backward (aft) pitching, left and right rolling (roll) of the helicopter. It is used by the pilot's right hand.
Collective- this controls the helicopters altitude. It is called collective because (I'm winging it here) it collectively adds pitch to all the rotor blades at once. The collective is used by the pilot's left hand. This will make more sense later.
Pedals- the pedals control the yaw of the helicopter or the left and right pointing of the nose with respect to the horizon. The pedals control the pitch of the tail rotor that counteracts the torque created by the main rotor. They also are used for directional control of the helicopter during hover- a.k.a. where the nose points. They are located at the pilot's feet.
Rotor Side Controls
The rotor side controls translate what the pilot does into something the rotor can do. This is where the swash plate comes in. There are MANY different ways to implement a swash plate. The swash plate physically translates the up and down motion of the controls (non rotating) to a rotating type control. The basic design of all swash plates is the same. The bottom half does not rotate and the top half does. The picture to the right shows the two parts.
The examples you'll see here are from a model helicopter the Kyosho Nexus 30. This swash plate uses a 180 degree swash plate mix (in radio controlled helis we call it swash plate mixing). This means there are two control arms spaced 180 degrees apart. Other mixing options are 120 degrees (three control arms) and 90 degrees (four control arms). Each mix method has advantages and disadvantages. Full size helicopters normally utilize 90 degree mixing because it provides more strenght to the swash plate and is more stable given the weight of full size helicopters.
The cyclic and collective controls are connected to the swash plate. The swash plate is a double sided "plate". The bottom side is connected to the pilots controls and the top side is directly connected to the rotors. The swash plate moves in all directions to allow control of the different pitches of the main rotor.