Both sides failed to claim a decisive victory; the German forces occupied 90% of the city at their peak, but never managed to completely eliminate resistance, while the Soviet armies kept sending reinforcements. Both sides indiscriminately air-bombed and howitzer-shelled the city; in fact, despite the freezing temperature of -30 centigrade, air within the city was hot from the constant gunpowder explosions. Eventually, the fighting degenerated into a bloody meatgrinding house-to-house warfare, dubbed by the Germans as rat-war.
The Soviet reserves eventually broke through the outer defences, and encircled the city, trapping the Germans in a pocket. The German commander, von Paulus, repeately argued before Hitler that a breakthrough retreat was vital to the German survival, but the madman Hitler prohibited retreat, which eventually caused the entire German army group to collapse.
At over a million casualties from each side, the battle of Stalingrad was perhaps the bloodiest battle of the war. It was especially devastating to the Germans, however, as they lost their best equipment, and unlike the Soviets, did not have the reserve manpower to replace the terrible losses. The battle of Stalingrad is considered the most important turning point on the Eastern front, and quite possibly the single most important battle of the war.
Maybe this is why the Russians call WWII "The Great Patriotic War".