Much as atomic theory and quantum mechanics underlie all of physics, but were only really discovered in the last century or so, set theory only dates back to the late 19th century and the work of Georg Cantor. The set is the most basic concept in all of mathematics, and set theory has been described as the foundation of mathematics. Counting, the basis of arithmetic, is a concept that arises from set theory, which distinguishes between countable and uncountable kinds of sets and studies their properties.
Abstract geometric concepts easily lend themselves to description by way of sets. The set of all squares, for example, is a subset of the set of all rectangles, which in turn is a subset of the set of all quadrilaterals, itself a subset of all polygons. The set of squares is also an intersection of the set of rectangles and the set of equilateral quadrilaterals; there are equilateral quadrilaterals (rhombi) that are not rectangles, and rectangles that are not equilateral, but squares belong to both sets.
A two-dimensional plane is identical with the set of all points that lie in that plane. Understood this way, other mathematical concepts become clear as sets themselves. A line or a curve, for example, is the subset of the plane consisting of all those points which have a particular mathematical relationship between their X and Y coordinates. The distance between two points is a measure of the size of the subset of points lying on a line between those two points. Geometric constructions by compass and straightedge amount to finding the intersections between two or more sets, and making inferences about the relationships between points and sets of points.
In the classical, Euclidean geometry dating back to the ancient Greeks, subsets of the plane could be described mathematically as lines and arcs. With the Cartesian coordinate system, a new way of describing curves as mathematical formulae became possible. The elegant curves of conic sections can be extrapolated after calculating only a few points by hand. Now, computers have expanded this power yet again, allowing mathematicians to visualize sets of numbers with startling complexity and beauty by calculating and displaying each point on the plane, pixel by pixel. The endless depth of the famous Mandelbrot set is a spectacular application of set theory and geometry.