El Alamein, the turning point of WW2
The battle at El Alamein in October 1942 between the Allied forces and the Axis forces is widely considered to be pivotal to the direction of the second world war. Churchill said "before Alamein we never had a victory. After Alamein we never had a defeat."
1942 was a particularly tough time for the Allies, having lost battles and ground on the Eastern Front (Russia) and in the Desert War (North Africa). Japan had recently entered the affray by way of Pearl Harbour in December 1941, so the war had become truly world wide. The Allied forces had been defeated in North Africa during 1942, and Churchill was facing political pressure at home. The Australian and New Zealand governments were losing confidence in the British handling of the war and growing concerned about the Japanese threat on their doorsteps. Japan's entry had forced the entry of the United States, which the Australians saw as having a common pacific interest and of being more strategically important than a distant Britain.
In North Africa, General Claude Auchinleck had led the Allied forces against the Axis forces, since mid 1941, with Lieutenant General Neil Richie leading the 8th Army. A string of failed battles against General Erwin Rommel culminated in the loss of Tobruk in June 1942. The concerned Churchill, under much pressure at home, paid Egypt a visit in July, subsequently replacing Auchinleck with General Harold Alexander and Richie with General Bernard Montgomery. Montgomery organised the forces, and led a successful battle in October 1942, culminating in a recapture of Tobruk in November 1942 and the withdrawal of Axis forces into Tunisia. The Axis forces faced further defeat, and withdrew across early 1943 followed by Rommel's departure for health reasons in March 1943, before complete defeat by the Allies (now joined by United States forces) during May 1943.
The combined Allied forces (Commonwealth, United States and a recently "turned" Italy) went on to further victories in the Mediterranean and across Europe, ultimately to win the war. Australian forces (i.e. the 9th Division) and New Zealand forces were taken from North Africa to South East Asia to work with United States forces in containing Japanese aggression and protecting Australia and New Zealand.
The campaign in North Africa has been singled out particularly for its notable leaders, especially Montgomery and Rommel, but also because it received much media attention, and thus provided key imagery about the progression of the war. Unlike events in Europe, North Africa was not tainted by the actions of the German SS and Jewish "cleansing". Arguably, the arrival of supporting United States forces into Egypt in late 1942 would have ultimately resulted in victory for the Allies, even had El Alamein itself failed, but the victory was well timed and necessary for overall morale.
Montgomery, "Monty", and the 8th Army
Unlike any of his predecessors, whether in North Africa or elsewhere, General Bernard Montgomery brought to the army a regime of professional training and planning. The British Army was generally inflexible in its approach: battles were planned to detail and then executed, without much ingrained initiative on the battlefield. This is why the better trained Axis forces, despite lower numbers and lower armaments, were able to fend off the Allies.
Montgomery realised that the 8th Army was not in shape for a battle, and any premature action would result in further losses and lowering of morale. His leadership style was impressive: he used powerful language and imagery, and cultivated symbolism and the media to lift morale and engage the hearts and minds of all soldiers. It worked at El Alamein in the decisive final battle, and beyond: the 8th Army having low casualty rates and a certain loyalty between its members.
Montgomery later led the 8th Army through Italy, and then commanded all ground forces in Operation Overlord (the D-Day invasion, etc). He remains a controversial figure for his judgmental comments and lack of diplomatic tact.
Further details: Bernard Law Montgomery - War in the Desert.
Rommel, "The Desert Fox", and Panzerarmee Afrika
It was generally, and incorrectly, believed that the Axis were a "stodgy" army, yet this myth was plainly dispelled by Field Marshal Erwin Rommel and his command of the Panzerarmee Afrika (the collection of German and Italian troops serving in North Africa). These soldiers had the benefit of many years of pre-war training, in which delegation and adaptive initiative were ingrained: particular objectives were imparted by command, and subordinates were expected to meet these objectives using their own initiative. Soldiers were trained to the level of their superiors, allowing them to take over during battle in the event of the superior being lost.
Rommel exemplified the master professional soldier, and commanded respect from Allied and Axis powers alike. Despite being poor on strategy, he was a brilliant tactician, who could and would adapt plans during the course of a battle, for which he attracted the name "The Desert Fox". As a very hands-on soldier, he could be found performing the duties of the average soldiers, such as offloading trucks. He would actively drive into and around the battlefield during action, observing, communicating and commanding. Rommel continually, and scholars tend to agree, berated German high command for supply and logistical failures: he was only ever defeated for these reasons.
Further details: Erwin Rommel - War in the Desert.