Currently under time constraints so my answer will not go to a huge level of detail.

Whilst the Germans did manage to level large parts of the city, this actually served to make it even harder to take ground, as it provided more ground level cover for the defenders. Urban fighting as a general rule favours the defensive side heavily, and stiff resistance mounted by the Soviets made it a major struggle for the Germans. Another thing to take into consideration is the issue of supplies. Partisan raids in the rear took a toll on German supply routes, and though not a decisive factor, did have an influence on the battle. One of the major factors that favoured the Germans in the invasion of the USSR was overwhelming air support, and their armour tactics. In Stalingrad, the response to this was to "hug" the German lines as close as possible, nullifying ground strafing/bombing, due to the risk of friendly fire. Furthermore, with so much of the battleground being rubble and collapsed buildings, armour was particularly susceptible to infantry attacks.

Stalingrad was one of the most successful examples of Soviet deception counter attacks. Whilst they used the bare minimum amount of troops to hold onto Stalingrad, convincing in part that they were on the verge of defeat, they amassed as much armour, planes and infantry behind the lines. Once they were satisfied with the build up, they smashed around the beleaguered German forces. Though the Germans attempted to reinforce and supply through the air, the Luftwaffe was simply not capable of this. The counter-attack relied on the Germans believing htat they were safe in their possession of the majority of Stalingrad, and an apparent imminent collapse of the defenders. There appears to be a persuasive view in the general public that the plucky Soviet defenders pushed out from within the city, rather then the massive encirclement that actually took place. Probably the fatal decision for the Germans was not to attempt a breakout immediately, but rather to attempt to break the encirclement and continue the siege. The commander of the German 6th Army Paulus has been heavily criticised for not taking action against the encirclement, but rather holding on for as long as possible as per Hitler's instructions.

Edit. Ignore the part about the immediate breakout and have a look at nickik's much more accurate response below. Because nickik is entirely correct, I was lazy and misinterpreted some of the source material.

Sources.

M Barbier, A Weist, "Infantry Warfare: The Theory and Practice of Infantry Combat in the 20th Century"

A Beevor "Stalingrad"

The Battle of Stalingrad ended poorly for many reasons, despite such a strong beginning for the Wehrmacht. Some of these are mistakes made by the commanders of the primary German formation, the 6th Army, and others are due to the Soviet Union’s 62nd Army, and its positive strengths, as well as those of its leader, General Zhukov.

One of these reasons was the string of bad choices made by the German High Command over the year-long siege. First, and possibly most importantly as a basis, there was a crucial mistake the Germans made on the first day of the battle. The bombing of the city, which required that vast quantities of Wehrmacht bombers and Stukas be drawn from all areas of the front, seemed a good idea initially: bomb out the civilians and possible partisans, and any garrison soldiers. However, this would later become one of the main reasons the city could be held by the Red Army. The completely destroyed ruins meant some areas were near impassable even by foot, created conditions where long range fighting was difficult and battles often became close range and even hand to hand, and any armoured fighting was instantly pitched in favour of the infantry, should they possess anti-armour weapons and equipment (the anti-tank rifle, which looked much like a large sniper, was used to great effect in these conditions). Another mistake then, which ties into this point, is that the commanders of 6th Army kept their panzers fighting in the city, which was not only wasteful, but can be said to be one of the contributing factors to the eventual encirclement of the 6th Army, as they were unavailable for recall, counteroffensive, or repositioning to break out of the trap engaged on 19th November 1942.

The 6th Army was also stretched precariously along the Don River. The single army held the entire stretch of river for over 150 miles, and had only a handful of divisions in reserve. Whilst this is alone a problem, the main issue is that the flanks of the 6th were protected by two Romanian armies in the north and south, as well as an Italian in the north. When the time for the Red Army’s offensive came, her attacking armies, in which the 62nd was not included, (1st Guards Army, 5th Tank Army, 21st Army, 65th Army, 24th Army, 66th Army, 64th Army, 57th Army, 51st Army) hit hard around the flanks, where the Romanian troops had not the proper equipment, nor full-strength division, not the morale; and then there were the Italians, who were Italian in all military sense, and thus had little equipment or morale. In both allied armies, the fighting men were also looked upon with disdain from their commanding officers, which Erwin Rommel also witnessed in Africa, stating that “it is as though they are two different species” (paraphrasing) – this did not encourage their soldiers to fight well. The Red Army armoured formations smashed through these Axis-allied armies with little issue (though being early-war Red Army they were still inexperienced forces and suffered surprising casualties, considering the calibre of the 6th Army’s flanks). The fact that, as earlier stated, there were no German response armoured divisions, and few reinforcing German formations, plus a hesitance to act on behalf of the German command, all meant that there was a quick, if costly, assault and encirclement carried out by a large Russian force on an overextended Axis formation, which, in the end, could simply not break out effectively (other reasons out with the boundaries of the question exist as to why the formation capitulated and was completely destroyed in the end).

The shear tenacity and sometimes brilliance of Red Army soldiers cannot be overlooked though. The resolve of their men was impeccable for much of the battle, though many still did dessert under incredibly stressful conditions, and many were executed for this. Red Army tactics also evolved to fight in a war which mirrored WW1 in brutality but was almost purely urban – hand to hand combat; eight-man fireteams armed with knives, SMGs, and snipers; the use of bombed out buildings as key fortresses and choke points; traversing sewers; hugging the enemy lines (the front line was never more than 100 yards from the Germans’, so that the German artillery had to be very particular and careful in danger of friendly fire); jury-rigged weapons and equipment; traps and contraptions MacGuyver would be proud of (AA-guns made of wood and stolen German MGs, which were adept at taking down and scaring off Stukas); and of course, pure resolve. The battles were so intense that they were known as Rattenkrieg (rat war) to the Wehrmacht soldiers. The tenacity of the defending 62nd Army at Stalingrad, and their ability to hold onto the west bank of the Volga river (by the skin of their teeth for the most part) as well as simply having the bodies to throw at the problem ensured that the Red Army continued to hold onto the key city until the two Red Army operations, Little Saturn (cancelled in favour of operation Uranus) and Uranus, could be enacted to relieve and recapture the city proper.

Finally, the aid of the Allies, whilst not often acknowledged, gave the Red Army the aid it needed. Actual fighting equipment give through the lend lease was considered inferior and ineffective by the Red Army, the most famous being that they thought the M4 Sherman was utter tripe, but Allied grain, meat, and Studebaker, Ford, Willys, et cetera, trucks and jeeps made all the difference in the capability to fight longer and achieve greater strategic reach. The trucks especially were useful, and over 600,000 were delivered over the course of the war. They were used to carry artillery pieces across the field quickly, and proved invaluable in the offensive at Stalingrad, as they did in so many battles.

Sources:

Anthony Beevor’s ‘Stalingrad’.

Alan Clark’s ‘Barbarossa’.

BBC’s ‘The World At War 1973, Episode 09 – Stalingrad’.