The Book of Exodus is the second book of the Pentateuch in the Old Testament of the Bible. Exodus documents the timeframe of the tribes of Israel being enslaved by Egypt up until their early establishment as an independent nation and begins the documentation of Israel's laws. The Book of Exodus is named after The Exodus, the event in which Israel "exited" Egypt, by way of the Red Sea.
All of the events of Exodus took place during Moses lifetime. The only exception to this is the very first chapter which continues the account of Hebrew settlement and enslavement in Egypt. Many events in the book, like Moses infant account or killing of an Egyptian would have been recorded many decades later. It is likely Moses did not begin to write down the events in the book until he was on Mount Sinai- at the age of eighty years old.