Prior appropriation water rights is a way that people make sure that water is used fairly. In some places there isn't enough water for everyone, so people have to decide who gets the water and when.
With prior appropriation water rights, people decide who gets the water first by who used the water first. Whoever started using the water first gets to keep using it and other people have to wait or find a different source of water. It's kind of like if you have five pencils, and the first person to grab one gets it and the other four have to wait until the first person is done with it.