Permit means to allow. When you permit your brother to come into your room, you let him in. (You can always kick him back out if he messes with your stuff.)
To permit means to let someone do something. It comes from the Latin permittere which means "give up, allow to pass through." You might feel like you "gave up" when you permitted your brother to enter your room. As a noun, a permit is a document that lets you do something — like when you get a driver's permit to learn to drive, or a gun permit to carry a gun, or even a construction permit to build an addition to your house.