Before the operation, there was general disagreement on where to proceed. General George Patton favoured an attack, with his troops, through Alsace-Lorraine on the old German border and British Field Marshal Montgomery favoured an attack, with his troops, through the Netherlands. Either attack would have had to cross the Rhine. The problem for the allies was that as not all the channel ports had been cleared and restored to use, enough supplies for only one offensive operation were available at that time.

I don't have full details of the point in which the order was decided, but more or less, historiography often puts this situation down to a huge personal rivalry between Patton and Montgomery. I considered saying more here, but I thought I would instead make a disclaimer that I have a very negative view of Patton and just leave it at that.

Montgomery's idea was selected. It was unusually ambitious for him, and it went along these lines: there were three river lines that he needed to cross to enter Germany, with the Rhine being the final. They needed to capture the bridges on these rivers (and the positions ahead of the crossing points) intact to bring over a lot of materiel. So they would drop paratroopers - American 101st Airborne Division at the Son Bridge, American 81st Airborne Division at the Nijmegen Bridge (Grave Bridge), and British 1st Parachute Division assisted by Polish 1st Airborne Brigade at the final bridge, Arnhem, which crossed the Rhine but was in (former) Dutch territory. Once all the bridges were captured, the British XXX Corps, with its heavy armoured and mechanised units, would rush across.

The plan didn't work. The Germans blew the bridge at Son before the American paratroops could capture it, and XXX Corps was not expecting this. The Americans did not take Nijmegen in time and the British had to withdraw from Arnhem before XXX Corps arrived as the pressure from Germans there was too strong. The road that XXX Corps had to move its armoured units down was only a single road with two lanes (or around that size) and it was often held up by stubborn German resistance. Additionally they had to deploy pontoon bridges to cross at the Son after it was blown up, and these bridges were far down the route of march, so the whole column had to be stalled to bring them up. By the time XXX Corps was at Arnhem Bridge, which the British did manage to take, the Germans had already ejected the British paratroops. The landing had not gone well and the British troops were scattered across the countryside and could not fight as one cohesive Division.

It's considered a failure because Arnhem Bridge wasn't captured and no route over the Rhine was made before winter of 1944. Only two of the three objectives were secured leading to the infamous quote "A Bridge Too Far"; however this was actually a quote made before the operation although it was then used as the name of the book and the eponymous, and popular, film. On the other hand, you may conclude that in eight days, (more like six), the allied troops advanced about 60 miles into German held territory, when the Germans actually outnumbered them, that this is actually not a strategic defeat at all. That was 60 miles of very defensible terrain which could have been put to good use by the Germans later on.

I am restraining myself from further comment on that topic, because at the moment, as you could be aware, Anthony Beevor's Arnhem book has recently been released. I haven't read it (I will at some point) but from what I hear it's more or less a condemnation of Montgomery. If you're going to read it you should read also a counter-balancing book of the operation.

Edit: Whatever - I will comment. It took Patton approximately twice as long to recover the same amount of ground in Normandy, when he had total air superiority and multiple possible routes of advance over no less difficult ground. There is nothing to suggest that if Eisenhower had chosen the right-front that Patton would have done anything more impressive than Market Garden.