The first day of the Yom Kippur war was on October 6th of 1973.
Also known as the 1973 Arab Israeli war, the Yom Kippur war started when Egyptian and Syrian forces launched a coordinated attack on Israel during Yom Kippur, the holiest day on the Jewish calendar. This took the Israeli Defense forces completely by surprise, and some of the Israeli soldiers had vacated their as posts in observance of Yom Kippur. What's more, that year, Yom Kippur coincided with the Muslim holy month of Ramadan.
Most of the fighting during the Yom Kippur war had taken place on the Sinai and Golan Heights. These territories had been occupied by Israel since the Six Day War of 1967. Armed hostilities continued thereafter, and eventually escalated to the War of Attrition, which was an attempt to wear down the Israeli position by way of long term pressure. Just as the lid of the international pressure cooker was ready to pop, a ceasefire was signed in August of 1970.
Only about a month later, in September of 1970, Egyptian President Gamar Abdel Nasser died, and Anwar Sadat succeeded him as the President of Egypt.
The Yom Kippur war was part of the ongoing (forever, it seems) Arab Israeli conflict, which led to many battles since 1948, the year of the founding of Israel. During the six-day war of 1967, which the other side had started, Israel had captured the Sinai Peninsula in Egypt, as well as about half of Syria's Golan Heights. These West Bank territories had been held by Jordan since 1948. The Yom Kippur war was an attempt by Egypt and Syria to get the land back.
The United States and the Soviet Union began massive resupply efforts to their respective allies, bringing the Yom Kippur war to the brink of nuclear disaster.
The intention of Egyptian President Anwar Sadat was to reopen the Suez Canal. In fact, the 1973 Yom Kippur war had started with the crossing of the Suez Canal by many thousand Egyptian troops. Both Syrian and Egyptian forces had crossed ceasefire lines to infiltrate the Sinai Peninsula.
Neither the Syrians nor the Egyptians actually planned to destroy Israel, but the people of Israel were not convinced of their intentions. Israel fought back ferociously for about for nineteen days, pushing the Syrian forces back behind the lines of ceasefire. Sadat ordered Egyptian soldiers to go back to the offensive position, but the Egyptians were swiftly repelled.
The Israelis then counter attacked at the seam between two Egyptian armies, crossed the Suez Canal into Egypt, and slowly advanced to the city of Suez. It took the Israelis about a week to do this, causing much bloodshed on both sides of the military equation. The battle went on until October 25th, when Israel finally emerged victorious.
In response to a United Nations inquiry, the Syrians and the Egyptians blamed each other for crossing ceasefire lines to begin with. At later peace talks, Israel basically agreed to return the land in exchange for peaceful relations.