Australians fought in many World War II campaigns - North Africa, Crete, Greece, Italy and New Guinea.
In only one of those campaigns did they fight under solely Australian command against a direct threat to Australia itself - the Battle of the Kokoda Track in New Guinea against the invading Japanese, between August and November, 1942. The Japanese had landed in New Guinea in March 1942 and in July started an advance across the island with the intention of taking the capital, Port Moresby.
The advance took place along a single track (the Kokoda Track that gave the campaign its name) though the mountains of the 7,000 ft-high Owen Stanley Range: Hot, humid days with intensely cold nights, torrential rainfall and the risk of endemic tropical diseases such as malaria make it a challenge to walk (Wikipedia).
The conditions were the same for both armies. But the Japanese were perhaps the toughest soldiers of World War II - disciplined, aggressive and suicidally brave. Opposing them were the inexperienced "chockos" - chocolate soldiers of Australia's militia, who had never seen battle.
Forces were small compared to European engagements - in the Pacific War, battalions counted as divisions in the West. The Japanese had 13,500 men. Perhaps only a few thousand Australians were available for the front line at the height of the campaign. At least, the Australians had command of the air and support from Port Moresby, plus shorter supply lines as they withdrew.
The campaign consisted of a series of engagements as the Australians tried repeatedly to halt the Japanese advance, but always found themselves outflanked by superior forces. Animated accounts of the campaign can be found here. The Kokoda Track | Australians in World War II | The Pacific War | About the Kokoda Track: 1942 and Today
The Japanese did eventually reach a point where they could see the lights of Port Moresby and the Coral Sea, but they were exhausted, hungry and at the end of their tether. Gradually, they had to withdraw in the face of counter-attacks from the reinforced Australians. Eventually, the Japanese landing place at Buna was captured by a joint Australian-US operation in March, 1943. Only 5,000 Japanese survived the Kokoda Campaign. Australian dead was about 700.
The campaign took place after the American victory at Midway (June 1942) and more or less at the same time as the Battle of Guadalcanal, so that the Australian Army and the "chockos" can claim an important role in turning the tide in the Pacific.
Kokoda Track campaign - Wikipedia, the free encyclopedia