Short answers, pulled from the story.
The Philippines campaign began on the 20th of October 1944, when the U.S. Sixth Army landed on the eastern coast of Leyte. Japanese forces had occupied the Philippines since 1942, and the invasion opened the campaign to drive them out.
American, Filipino, Australian, and Mexican forces took part in the Philippines campaign. Filipino guerrillas fought directly alongside U.S. Army units, Australia supplied Royal Australian Air Force and Navy support including a heavy cruiser, and Mexico's 201st Fighter Squadron flew air support.
The Battle of Leyte Gulf was a major naval engagement fought from the 23rd to the 26th of October 1944, during the Philippines campaign. It destroyed the remaining strength of the Imperial Japanese Navy, including four aircraft carriers, after Japan tried to attack the American landing force at Leyte.
Japanese forces in the Philippines were ordered to surrender by Tokyo on the 15th of August 1945, following the atomic bombings of Hiroshima and Nagasaki and the Soviet-Japanese War. Fighting on islands such as Mindanao and Luzon continued right up until that surrender.
American Army and Army Air Forces losses in the Philippines campaign totaled 13,884 killed and 48,541 wounded, for 62,425 total casualties. Japanese losses were far higher at 320,795 killed and 12,573 captured, and an estimated one million Filipino civilians died in the campaign.
The last known Japanese holdout from the Philippines campaign, Hiroo Onoda, did not surrender until 1974. He held out for decades in the mountains of Lubang Island near Mindoro.