The specification real simply means that we are considering a projective plane whose coordinates, if we were to calculate them, would all be real numbers. So now, what is the projective plane? Projective geometry is based on the Euclidean geometry that we all know and love, but it is more generalized. The projective plane can be thought of as a Euclidean plane with the addition of a line at infinity. The addition of the line at infinity closes the projective plane, giving it special properties that the Euclidean plane does not share.

• On the projective plane, any (every) two coplanar lines intersect at a point. (Parallel lines intersect at the point at infinity.)
• Similarly, any two planes in projective geometry intersect at a line. Parallel planes intersect at the line at infinity.
• A line on a Euclidean plane can be divided into two segments by "cutting" at one point. However, a line on a projective plane needs two distinct points to divide it into two segments. This is because a line on a projective plane meets itself at the point at infinity, and is therefore closed. It may be helpful to visualize the line as a circle -- a circle would need to be "cut" at two distinct points to form two segments.
• On a line on the projective plane, the order of points is cyclic, whereas for a line on a Euclidean plane, the order of points is serial. This is because a line on the projective plane is closed (as defined above), while a line on the Euclidean plane is open.

Links:
A page about the projective plane with helpful diagrams.

The Geometry Center's page about the real projective plane.

Source:"The Real Projective Plane" by H. S. M. Coxeter (Cambridge University Press, 1960)

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