Very roughly speaking, geometry is that part of mathematics that studies properties of figures. Often, the reasoning used in geometry itself is of geometric nature, i.e. one reasons with properties of figures (as say is done in classical Euclidean geometry). However, there is also the possibility of using algebraic reasoning (as is done in classical analytic geometry or, what is the same thing, Cartesian or coordinate geometry), combinatorial reasoning, analytic reasoning, and of course combinations of these different approaches.
In contemporary mathematics, the word ``figure'' can be interpreted very broadly, to mean, e.g., curves, surfaces, more general manifolds or topological spaces, algebraic varieties, or many other things besides. Also, the word ``space'' is used more often than the word ``figure'', as a description of the objects that geometry studies.
There are also many aspects of figures, or spaces, that can be studied. Classical Euclidean geometry concerned itself with what might be called metric properties of figures (i.e. distances, angles, areas, and so on). Classical projective geometry concerned itself with the study of properties invariant under general linear projections. Topology is (loosely speaking) the study of those properties of spaces that are invariant under arbitrary continuous distortions of their shape. In general, several of these different aspects of geometry might be combined in any particular investigation. In particular, although topology is less ancient than some other aspects of geometry, it plays a fundamental role in many contemporary geometric investigations, as well as being important as a study in its own right.
There are many techniques for studying geometry and topology. Classical methods of making constructions, computing intersections, measuring angles, and so on, can be used. These are enhanced by the use of more modern methods such as tensor analysis , the methods of algebraic topology (such as homology and cohomology groups, or homotopy groups), the exploitation of group actions, and many others.
Geometry and topology are important not just in their own right, but as tools for solving many different kinds of mathematical problems. Many questions that do not obviously involve geometry can be solved by using geometric methods. This is true for example in the theory of Diophantine equations, where geometric methods (often based on algebraic geometry) are a key tool. Also, investigationss in commutative algebra and group theory are often informed by geometric intiution (based say on the connections between rings and geometry provided by algebraic geometry, or the connections between groups and topology provided by the theory of the fundamental group). Certain problems in combinatorics may become simpler when interpreted geometrically or topologically. (Euler's famous solution of the Konigsberg bridge problem gives a simple example of a topological solution to a combinatorial problem.) There are many other examples of this phenomenon.