Leningrad, Moscow and Stalingrad were confronted by the Wehrmacht under different conditions than the cities taken the initial advance of Barbarossa.

It's worth noting here that cities like Brest, Zhitomir, Odessa, Riga, Kiev and Smolensk were the sites of pretty heavy fighting. In the cases of Moscow and Leningrad, these cities were confronted by German forces at the end of their logistical tether, and that were coming to terms with the cumulative effect of the casualties they had endured in the previous fighting. Leningrad was preceded by dogged Red Army defence of the Luga River, and Army Group North itself was spread out across a wide front, and did not posses the forces to take the second largest city in the Soviet Union by coup de main.

The situation at Moscow was even worse. Wehrmacht supply lines were in a shambles, exacerbated by inadequate aerial resupply assets, poor soviet roads and even poorer German transport, the difference between Soviet and European rail gauges, and deteriorating weather. combine this with the need to fight two major encirclement battles at Vyasma and Bryansk before they could so much as see Moscow on the horizon, steadily worsening weather, growing resistance on and behind the front lines, and a wide front to cover, it was asking a hell of a lot of Army Group Center to then take the largest city in the Soviet Union!

At Stalingrad, the Germans did control most of the city at one point, but the urban sprawl and ruins of the city posed a challenge that the Germans had never encountered before. Combine this with dogged resistance from the Red army, the close quarters nature of the fighting, which decreased the effectiveness of German artillery, armoured, and air support, and the need to guard the flanks of the 6th Army, which itself had no more than 1/3 of it's forces fighting for the city at any one time? It was a difficult task for the 6th army to tackle, and the mediocre generalship of Friedrich Paulus didn't exactly help things.

Stalingrad, Moscow, and Leningrad were three of the biggest cities in the Soviet Union by far, so clearly the Soviets were going to throw every possible resource into holding them. That alone is probably the most important reason they weren't captured.

The full story is has many books worth of material in it, but here are three other big reasons they didn't fall.

1. By this time in the war the Soviets were no longer caught off guard, and many fresh and fully equipped units were arriving at the front from the Eastern reaches of the USSR. They were also receiving lend-lease equipment from the United States.

2. Hitler's generals wanted him to commit far more troops for an attack on Moscow, which, if done well and early enough, could well have succeeded. Instead, he took troops away from those en route to Moscow and sent them to the Caucus to capture the oil reserves there.

3. At least with Leningrad and Stalingrad, the Nazis bombed each city into rubble, rubble which made urban engagements favor defenders by blocking roads and generally making life very difficult for armored vehicles, which were one of if not the most important part of the Nazi war machine.

If you are interested in learning more about these battles, I would very highly recommend Stalingrad by Antony Beevor, it is an excellent piece of WW2 history that is also very readable.