This is a question I am not really sure how to answer. So I put it to Sir Michael Atiyah after his Frontiers talk in Cardiff. In essence he told me that geometry is any mathematics that you can imagine as pictures in your head.
To me this is in fact a very satisfactory answer. Geometry a word that literally means “Earth Measurement” has developed far beyond its roots of measuring distances, examining solid shapes and the axioms of Euclid.
Another definition of geometry would be the study of spaces. Then we are left with the question of what is a space?
Classically, one thinks of spaces, say topological or vector spaces as sets of points with some other properties put upon them. The notion of a point seems deeply tied into the definition on a space.
This is actually not the case. For example all the information of a topological space is contained in the continuous functions on that space. Similar statements hold differentiable manifolds for example. Everything here is encoded in the smooth functions on a manifold.
This all started with the Gelfand representation theorem of C*-algebras, which states that “commutative C*-algebras are dual to locally compact Hausdorff spaces”. I won’t say anything about C*-algebras right now.
In short one can instead of studying the space itself one can study the functions on that space. More than this, one can take the attitude that the functions define the space. In this way you can think of the points as being a derived notion and not a fundamental one.
This then opens up the possibility of non-commutative geometries by thinking of non-commutative algebras as “if they were” the algebra of functions on some non-commutative space.
Also, there are other constructions found in algebraic geometry that are not set-theoretical. Ringed spaces and schemes for example.
So, back to the opening question. Geometry seems more like a way of thinking about problems and constructions in mathematics rather than a “stand-alone” topic. Though the way I would rather put it that all mathematics is really geometry!