A plane is a flat surface with no thickness.
Our world has three dimensions, but there are only two dimensions on a plane.
And it goes on forever.
It is actually hard to give a real example!
When we draw something on a flat piece of paper we are drawing on a plane ...
... except that the paper itself is not a plane, because it has thickness! And it should extend forever, too.
So the very top of a perfect piece of paper
Also, the top of a table, the floor and a whiteboard are all like a plane.
Point, Line, Plane and Solid
Plane vs Plain
In geometry a "plane" is a flat surface with no thickness.
But a "plain" is a treeless mostly flat expanse of land ... it is also flat, but not in the pure sense we use in geometry.
Both words have other meaning too: Plane can mean an airplane, a level, or a tool for cutting things flat; Plain can mean without special things, or well understood.
Imagine you lived in a two-dimensional world. You could travel around, visit friends, but nothing in your world would have height.
You could measure distances and angles.
You could travel fast or slow. You could go forward, backwards or sideways. You could move in straight lines, circles, or anything so long as you never go up or down.
What would life be like living on a plane?