Prestige will reset your mine so you play it from the beginning, but it will have a much higher multiplier which means you will earn a lot more money when upgrading the mine again. You can Prestige a mine six times in total, each time you will get an even higher multiplier.
To Prestige a mine, you need to spend some Cash. The cost is displayed on the Prestige pop-up each time, so make sure that you have enough Cash left to quickly get back to where you were before.
When you Prestige, you will keep the rest of your Cash (after paying the Prestige fee), plus your Super Cash and other mines.