What is synthetic geometry?

It´s geometry without metric (ruler, protractor, scales etc). Just with pure geometrical contents.
Ex.: questions about planes or lines intersecting points, lines intersecting planes etc are incidence synthetic geometrical questions.
Parts of the Elements of Euclid are synthetic. Hilbert's axioms of Euclidean Geometry are synthetic because you don't need to measure segments or angles, and congruence is a primitive relation.
Birkhoff´s axioms are not synthetic because distance, scale and real numbers belongs to the axioms. You have metric Geometry.