The Pythagorean Theorem (a2 + b2 = c2) states that the sum of the square of the lengths of two sides of a right triangle equals the square of the hypotenuse side (the side which is opposite the right angle). The applications of the Pythagorean Theorem are immense, as it is used to determine distances in the real world, including measuring objects in space. As long as you know three points are laid out in a right triangle, and you know the length of two of the sides, you can always determine the length of the third side.
Pythagoras was also the first to expound a theory of dimensions. Specifically, he progressed from a point (zero dimensions) until it became four points and a tetrahedron, which is three-dimensions. The applications of this are seen all throughout geometry; the creation of polygons in coordinate space comes from Pythagoras's conception of dimensions.
Pythagoras was much more than a mathematician; he was also a philosopher who believed that the universe itself was merely an expression of numbers, with 10 being the "perfect" number. This theory would be applied and expanded by Plato, and his concept of the perfect shapes and ideals. Specifically, Plato used Pythagoras to show that our world is merely a shadow of the "perfect" world, and numbers were proof of this. Pythagoras's philosophy would later be applied by Kant, Spinoza and other Western thinkers.
Pythagoras also was a leader in a religious cult that believed in the transmigration of souls. Pythagoras believed that only pure souls could actually transcend beyond the physical realm, and thus most human beings are reincarnated into other animals after they die. As an application of this belief, Pythagoras and his followers were strict vegetarians because they believed they would be eating a conscious being if they ate meat.