A projective plane is a projective space of dimension 2. However, projective planes over fields (or more generally division rings or ternary rings?) can be characterized axiomatically with a small list of “synthetic” axioms.
A projective plane consists of
satisfying the following axioms:
Some further axioms which may be added are:
the diagonal points of a complete quandrangle? are not collinear (“Fano’s axiom”)
if the lines joining corresponding vertices of two triangles are concurrent, then the intersections of corresponding sides are collinear (Desargues' theorem)
if the vertices of a hexagon lie alternately on two lines, then the intersections of opposite sides are collinear (Pappus' theorem?)
Of these, Pappus’ theorem implies Desargues’ theorem.
For any field $F$, we can construct the projective plane $F P^2$ in several ways:
The resulting plane satisfies Pappus’ theorem, hence also Desargues’ theorem. It satisfies Fano’s axiom iff the characteristic of $F$ is $\neq 2$.
These methods also work for any division ring, such as the quaternions. In this case, Pappus’ theorem doesn’t hold, but Desargues’ theorem still does.
The octonions are not associative and this breaks the first two methods, but the latter two can be made to work, resulting in the Cayley plane. In this case Desargues’ theorem does not hold in general, but there are special cases of it which do (corresponding to the fact that the octonions are “alternative”).
In any projective plane, we can define a “scalar” to be an ordered set of four collinear points $(A,B,C,D)$ of which no more than two are equal. Two scalars are considered equal if they are projectively related, i.e. the four lines joining corresponding points are concurrent. If Desargues’ theorem holds, we can define addition and multiplication on the scalars (omitting one of them that acts like $\infty$) making them into a division ring $F$ such that our plane is isomorphic to $F P^2$. The ring $F$ is commutative (hence a field) iff Pappus’ theorem also holds, and has characteristic $\neq 2$ iff Fano’s axiom holds.
If Desargues’ theorem fails, then we can still construct a sort of algebraic structure on the scalars, called a ternary ring?, which suffices to reconstruct our plane. However, distinct ternary rings can give rise to isomorphic projective planes, in contrast to the situation for fields and division rings.