So imagine you have a toy world with little toy people and toy buildings. In a persistent world, this toy world doesn't go away when you put it back in the toy box. It stays exactly the same way you left it, so when you come back to play with it, everything is still there just like you remember it.
In a video game, a persistent world is like this toy world, but on a computer. This means that even when you turn off your game and come back tomorrow, everything is just as you left it. The world is always "on," just like your toy world.
This means that other players can join the world anytime and they will see everything exactly as you did. They can build things in it, they can talk with the toy people in it, and they can explore whatever they want.
So, in a persistent world, everything in the game stays the same even if you turn it off and come back later, and other players can play in the same world with you.