Short answers, pulled from the story.
The Western Desert campaign began on the 11th of June 1940, when Italy declared war and hostilities commenced on the Egypt-Libya frontier. It ended in May 1943 when the remaining Axis forces surrendered in Tunisia.
Operation Compass was a British offensive that began on the 9th of December 1940, originally planned as a raid of four to five days. It destroyed the Italian 10th Army entirely, capturing 133,298 prisoners, 420 tanks and 845 guns over two months of fighting.
Axis defeat was fundamentally caused by an insoluble supply problem. The small size of Libya's ports limited how much materiel could be unloaded, and 35 to 50 per cent of fuel deliveries were consumed transporting the remainder to the front. Rommel himself wrote that Axis supply difficulties, relative to those of the British, determined the course of the military campaign.
Rommel commanded the Afrika Korps and later Panzerarmee Afrika from early 1941. He reversed the British conquest of Cyrenaica by April 1941, besieged Tobruk, and by June 1942 had driven the Eighth Army back to El Alamein, just 70 miles from Alexandria. He was defeated at the Second Battle of El Alamein in October-November 1942 and left Africa in March 1943.
Tobruk was the only deep-water port between Tripoli and Alexandria and changed hands multiple times. The 9th Australian Division held it under siege for nine months in 1941. It fell to the Axis on the 21st of June 1942 when 35,000 Eighth Army troops surrendered. The Eighth Army retook it on the 13th of November 1942 after the Second Battle of El Alamein.
When the Eighth Army offensive began on the 23rd of October 1942, it had 195,000 men and 1,029 tanks. The Panzerarmee had 104,000 men and 496 tanks, but only 180 miles' worth of fuel per vehicle. By the 2nd of November the Panzerarmee was down to 32 German and 120 Italian tanks, and on the 4th of November the Eighth Army broke through.