Short answers, pulled from the story.
The Budapest offensive ran from the 29th of October 1944 to the 13th of February 1945. The offensive lasted roughly three and a half months and ended with the fall of the Hungarian capital to Soviet and Romanian forces.
Approximately 79,000 German and Hungarian troops were encircled inside the Budapest pocket in late December 1944. When the garrison surrendered on the 13th of February 1945, fewer than 1,000 of those defenders escaped death or captivity.
Rodion Malinovsky commanded the 2nd Ukrainian Front, which led the first two offensive periods in October and November 1944. Fyodor Tolbukhin commanded the 3rd Ukrainian Front, which arrived after the liberation of Belgrade and helped complete the encirclement of the city.
Operation Spring Awakening was a German counteroffensive launched in the Lake Balaton area in March 1945. Its goals were to protect remaining Axis oil-producing regions and to retake Budapest, but neither objective was achieved.
Soviet claims recorded 49,000 German and Hungarian soldiers killed, 110,000 captured, and 269 tanks destroyed during the Budapest offensive.
The fall of Budapest on the 13th of February 1945 destroyed the main forces of Army Group South and left the routes toward Vienna, Czechoslovakia, and southern Germany open to Soviet advance. The offensive was later listed in Soviet propaganda as one of Stalin's ten blows, the major sequence of operations that reshaped the Eastern Front in 1944-1945.