The Aharonov–Bohm effect is like a magic trick. Imagine that you have a container that you can look through, but where nothing else can enter or leave. Inside this container you have some electrically charged particles, like tiny balls with positive and negative charges.
When you move the charged particles around inside the container, they create a magnetic field. The Aharonov–Bohm effect says that even though you cannot see the magnetic field (it's inside the container and you can't look in) it can still affect things outside the container.
For example, if you take a neutral particle (like a tiny uncharged ball) and move it around outside the container, the magnetic field inside the container can make the neutral particle do weird things. It's like the magnetic field inside the container is doing a magic trick on the neutral particle outside.