- Given any two distinct points, there is exactly one line incident with both of them.
- Given any two distinct lines, there is exactly one point incident with both of them.
- There are four points such that no line is incident with more than two of them.

Note that a projective plane is an abstract mathematical concept, so the "lines" need not be anything resembling ordinary lines, nor need the "points" resemble ordinary points.

Consider a sphere, and let the great circles of the sphere be "lines",
and let pairs of antipodal points be "points".
This is the **real projective plane**.
It is easy to check that it obeys the rules required of projective planes:
any pair of distinct great circles meet at a pair of antipodal points,
and any two distinct pairs of antipodal points lie on a single great circle.
If we identify each point on the sphere with its antipodal point,
then we get a representation of the real projective plane in which
the "points" of the projective plane really are points.
The resulting surface, a two-dimensional compact non-orientable manifold, is a little hard to visualize,
because it cannot be embedded in 3-dimensional Euclidean space without intersecting itself. Three self-intersecting embeddings are Boy's surface, the Roman surface, and a sphere with a cross-cap.

It can be shown that a projective plane has the same number of lines as
it has points.
This number can be infinite (as for the real projective plane)
or finite (as for the Fano plane).
A finite projective plane has *n*^{2} + *n* + 1 points,
where *n* is an integer called the *order* of the projective plane.
(The Fano plane therefore has order 2.)
For all known finite projective planes, the order is a prime power.
The existence of finite projective planes of other orders is an open question.
A projective plane of order *n* has *n* + 1 points on every line,
and *n* + 1 lines passing through every point,
and is therefore a Steiner S(2, *n*+1, *n*^{2}+*n*+1) system
(see Steiner system).

The definition of projective plane by incidence properties is something special to two dimensions: in general projective space is defined via linear algebra.