Two things that can't happen at the same time, or that completely contradict each other, are mutually exclusive. If you tell your dad you want to go hiking and see a movie on Saturday, he might tell you those are mutually exclusive choices: you can pick only one.
You could say that war and peace are mutually exclusive, since by definition peace includes the absence of war or fighting. Some things that seem like they're mutually exclusive (like sunshine and rain) aren't really — it's possible for it to rain when the sun is shining. In math and statistics, there are numerous examples of things that are mutually exclusive, and even formulas for determining whether or not they are.