A flat surface that is infinitely large and with zero thickness

Clearly, when you read the above definition, such a thing cannot possibly really exist.
Imagine a flat sheet of metal. Now make it infinitely large in both directions. This means that no matter how far you go, you never reach its edges.
Now imagine that it is so thin that it actually has no thickness at all.
In spite of this, it remains completely rigid and flat. This is the 'plane' in geometry.

It fits into a scheme that starts with a point, which has no dimensions and goes up through solids which have three dimensions:

It is difficult to draw planes, since the edges have to be drawn.
When you see a picture that represents a plane, always remember that it actually has no edges, and it
is infinitely large.

The plane has two dimensions: length and width. But since the plane is infinitely large, the length and width cannot be measured.

Just as a line is defined by two points, a plane is defined by three points. Given three points that are not
collinear,
there is just one plane that contains all three.

Parallel planes

You can think of parallel planes as sheets of cardboard one above the other with a gap between them.
Parallel planes are the same distance apart everywhere, and so they never touch.

Intersecting planes

If two planes are not parallel, then they will intersect (cross over) each other somewhere.
Two planes always
intersect at a
line, as shown on the right.

This is similar to the way two lines
intersect at a
point.

Naming of planes

Planes are usually named with a single upper case (capital) letter in a cursive script such as

Coordinate Geometry

In another branch of mathematics called coordinate geometry, points are located on the plane using their
coordinates - two numbers that show where the point is positioned. To achieve this, the plane
is thought to have two scales at right angles. Using a pair of numbers, any point on the plane can be uniquely described.