I know more about the British system used to detect German attacks, so I'll stick to that path.

Radar was not a new technology, direction finding (known as RDF) by radio had been around for quite a few years. In the 20s a man by the name of Robert Watson-Watt worked for the Meteorological Office, this was (still is) a weather forecasting group set up in the 1800s originally by the board of trade - primarily to try and avoid shipwrecks. By Watson-Watts time it was part of the Air Ministry.

Wattson-Watt was experimenting with the use of radio waves to detect thunderstorms with some success. n the late 20s the radio department of the Met Office was merged with a similar team from the Nation Physics Library to form the Radio Research Station, Watson-Watt was put in charge.

During WWI Germany had attacked using airships that dropped bombs, in the inter war years it was recognised that air defences needed to be improved, especially with the huge steps that aviation was taking, bomber aircraft from Europe would be in easy reach of the UK. All of this led Stanley Baldwin to issue his now famous quote "The bomber will always get through".

The Air Ministry realised that having fighter planes on patrols wouldn't be an effective defence, because you'd need so many. At the same time rumours began to circulate that the Nazis had a "death ray" that would use powerful radiowaves that would destroy anything in its path - including entire towns and cities. Watson-Watt was asked to investigate to see if such a weapon was plausible, after a few calculations he and a colleague Albert Wilkins summarised it wasn't. However Wilkins made a passing comment that it might be possible to detect aircraft by using radio waves.

Watson-Watt soon sent a memo entitled ""Detection and location of aircraft by radio methods" to the Air Ministry. They were curious but sceptical so asked Watson-Watt to prove his theory. He did, the Air Ministry began to get behind the idea and supply funding. In 1935 Watson-Watt was granted a patent for detecting an aircraft by radio waves. Watson-Watt and his team moved to Orford Ness to continue the work.

Soon a number of radar stations were planned all along the coast, there is a map here of all the stations; http://www.anti-aircraft.co.uk/chain_home_map.html Note each station comprised of a transmitter and receiver, usually a few hundred metres apart and a control office.

The thing that we must take note of is that the radar stations were only part of the system - they would detect incoming aircraft but how could that information be used effectively ? What they set up was a direct link between the radar stations, observers and the RAF command. So, the radar (or observers) reporting incoming aircraft, the control rooms had complete visibility of the enemy and their own forces and could quickly and accurately deploy their fighters to attack the enemy. Control rooms could even contact pilots in the air and re-direct them as necessary. The extra time afforded by the advance warning radar gave meant that fighter squadrons had precious extra minutes to get in the air. There is a longer description of how the control system worked here : http://www.raf.mod.uk/history/fightercontrolsystem.cfm

Here's a picture of one of the control stations, http://upload.wikimedia.org/wikipedia/en/thumb/b/bf/Plotting_Table.jpg/1024px-Plotting_Table.jpg. This was for 11 Group and is sited at RAF Uxbridge. The lists on the back wall would be used to show which squadrons the RAF had at which airfields and where they were (e.g. awaiting orders, in the air, in combat or being refuelled etc). WAAFs would stand around the map table, listening on headphones for information from pilots/airfields/radar stations/observers, they would use the pusher sticks to move markers about that denoted both RAF and enemy forces and their strength/height. The commanders would stand where the photographer is to get a view of the complete situation and give orders.

Radar was, really only half the answer, the control system (known as the Dowding System) made effective use of the information that radar gave.

At the height of the Battle Of Britain Churchill visited Uxbridge during a raid and watched what was going on, he asked Group 11's chief Keith Park "What reserves do we have?" Park scanned the room and replied "There are none", the RAF had thrown everything they had at the enemy. Two days later Hitler canceled the invasion of Britain.

Upon leaving Uxbridge Churchill said to Lord Ismay "Do not speak to me, I have never been so moved". A few minutes later he uttered the now famous phrase "Never in the field of human conflict has so much been owed by so many to so few". He re-used the phrase later in speeches.

There is a detailed explanation of how chain home radar stations worked here: http://www.radarpages.co.uk/mob/ch/chainhome.htm