So what is Air Traffic Control?

To get it straight right here - no, the friendly guy above is NOT an air traffic controller, but a marshaller, who guides airplanes in the direct vicinity of terminals at airports.
For some strange reason quite a few people seem to confuse these two professions on a routine basis.

Air traffic control roughly can be divided into two main areas - terminal control and approach/area control.
Common to both is the goal - to ensure the safe, orderly and expeditious flow of air traffic, in exactly that order.

The consequence of that goal is a very close cooperation of pilots and air traffic controllers to ensure safety of every flight, and every passenger.
The amazing growth of air traffic over recent decades, and the accompanying fact that almost everybody can now participate in it - either as pilot or passenger - would not be possible without that cooperation.

Terminal control

Or "yeah that's what I thought."

Tower controller at Munich, Germany. Source: DFS

Everyone knows the impressive structures at every airport known as control towers, and those beyond the stage of confusing air traffic controllers with marshallers usually even know that ATC sits behind those nice panorama windows.

Terminal controllers work traffic in the direct vicinity of an airport - not all airports though, as some smaller ones are un-controlled, with pilots ensuring their own separation.
From their vantage point in the tower they are both responsible for movement in the air - the so-called "control zone", and on the ground.

At least every commercial pilot starts his flight interacting with these controllers, getting a clearance for the flight plan he filed, receiving permission to start engines and receiving instructions/information up to the clearance for take-off.
On final approach, he again gets returned to terminal control, again receiving clearances and instructions, albeit in reversed order - first to land, then to taxi to his parking space.

The equipment these controllers use to ensure separation can differ slightly from airport to airport, ranging from completely visual without any aid by radar, to the highly sophisticated environment at large international airports, with air radar, ground radar but also still the human eyeball Mk.1
One of the most important pieces of equipment though is two-way communication radios, usually in the very high frequency (VHF) range, which is used to transmit instructions/information to pilots.

Approach/Area Control

Or "So these guys don't actually see airplanes?"

Area control center at Langen, Germany. Source: DFS

While the role of tower controllers usually is no problem to understand even for the uninitiated, center controllers seem to be a different matter.

Starting where we just left, with terminal control - once an airplane is airborne, it gets switched over to an approach controller.
These controllers serve the greater vicinity of an airport, usually to distances of around 60km (compared to usually 20-30km for terminal control).

Then, once you leave this defined airspace, you are taken care of by area controllers.

Using radar almost exclusively, these two types of controllers also ensure the separation of traffic, based on minimum distances which are directly derived from the accuracy of the radar involved - the more accurate, the closer airplanes may come to each other.

Since, generally speaking, long-range radars are less accurate than short-range radars, enroute controllers (covering vast distances) usually have to maintain a higher separation than approach controllers, usually about 9km (5 nautical miles) as compared to 5.5km (3 nautical miles).
Alternatively, they may use vertical separation, which normally is at least 300m (1000ft).

This separation is achieved by guiding the traffic via instructions, to alter their headings, change speed or climb/descend to a new altitude.

Just like the terminal controllers, approach and area controllers use radios to transmit their instructions to airplanes in their area.

One key difference comes from the fact that it is quite hard to keep a visual watch on airplanes 100km away at high altitudes: these controllers sit in (usually window-less) rooms, sometimes not even directly at airports.
And yes, it would be quite rare for them to actually see an airplane.

These little abstracts can only give a very short introduction into the highly complex system necessary to ensure that whenever you fly, you arrive at your destination safely.
A few more details can be found at this Wikipedia article.