Short answers, pulled from the story.
The Second Battle of El Alamein began on the 23rd of October 1942 and concluded with Axis retreat orders issued by Erwin Rommel on the night of the 3rd to the 4th of November 1942. The thirteen-day battle started under a full moon and ended after Operation Supercharge forced the Panzer Army Africa to withdraw westward.
Lieutenant-General Bernard Montgomery led the Eighth Army against Axis forces starting from his arrival in Egypt in August 1942. He replaced Lieutenant-General William Gott who was killed when his transport aircraft was shot down by Luftwaffe fighters before the offensive commenced.
Commonwealth forces executed Operation Bertram which included dummy pipelines, plywood tanks disguised as supply trucks, and waste material dumps to mislead Axis command about the timing and location of the assault. These measures allowed the Eighth Army to build up supplies unnoticed while confusing enemy intelligence regarding their true intentions.
Axis losses totaled approximately 25,000 men killed or wounded including 5,920 Italians plus 30,000 prisoners taken during the retreat phase. Italian official history records between 4,000 to 5,000 killed or missing along with 7,000 to 8,000 wounded and 17,000 prisoners.
Erwin Rommel ordered a retreat because he received no reply from Adolf Hitler allowing him to fall back on Fuka after realizing the situation was desperate. British blows had encircled key divisions like Ariete and Trento while Ultra intelligence exposed critical shortages in fuel and ammunition for Panzerarmee Afrika.