Long story short. The idea was to take some the bridges in the lower Rhine in The Netherlands. If that had happened, we would have advanced into the Ruhr Valley (or Ruhr Region/Ruhr District) bypassing most of the German army on the Western front. It also meant advancing into Germany's main Industrial area. The idea behind all that was to end the war by Christmas 1944.

The main reason it didn't work is because of a number of things. We took a lot of the objectives early on, but when we got to Arnhem bridge (The British and Polish) we came across a lot more Germans than we thought. What made it worse is that they literally dropped in on Walter Model who realised that they were being invaded. When he realised this he ordered the II Panzer Corps (9th and 10th Panzer Divisions) and ordered the 9th Panzers to go to Arnhem and the 10th to go to Nijmegen. They were veteran soldiers that had just come from Normandy. It ended up with the Germans on one side of the Bridge and the Brits and Polish on the other with both sides trying to take each others part. Eventually the British got surrounded and had to evacuate.

I don't know much about the American side, so you will have to wait for someone else to talk about it or look it up yourself.

Correct me if I'm wrong about anything (I'm not a historian)

Also if you have a lot of time to get rid of, watch the film http://en.wikipedia.org/wiki/A_Bridge_Too_Far_(film) it's really good.