The fundamental theorem of algebra is that if you include complex numbers, all polynomials have solutions. Complex numbers complete the number system. They are also useful in areas like describing the relation between current and voltage waves when they are out of phase. When figures are drawn on the complex plane, they are easy to rotate by simply multiplying the points of a complex number. An extension of complex numbers, called quaternions, are used to draw the figures in computer-generated images so they can easily be moved and rotated.
Cartesian coordinates are the invention of one man: Rene Descartes. Cartesian coordinates put a frame on Euclidean geometry and gave an address for every point in the plane. Using Cartesian coordinates, it is possible to turn equations and functions into pictures that display the roots of an equation and the places where a function is maximum and minimum. Cartesian coordinates also paved the way to the invention of calculus and made many mathematical concepts clearer by making them visual.
Group theory is a central concept of abstract algebra and is useful for finding techniques and proving theorems in very traditional mathematical arenas, like showing that there is no algebraic formula for solving fifth degree equations. Group theory is also useful in proving theorems in physics and other arenas that are hardly traditional at all, like finding solutions for a Rubik's cube. A group is any set with only one function defined and only a few restrictions on the function. The real power of group theory is that the set can be anything: rotations of a Rubik's cube, positions of quarks, spin of electrons, blood types or the rules of grammar.
Boolean algebra is an algebraic structure applied to logic. Algebra was invented to manipulate numbers, but Boolean algebra was invented to manipulate statements that can be true or false. Traditional algebra uses functions like addition and multiplication, but Boolean algebra uses functions like AND and OR and NOT. Algebraically, Boolean algebra is a "ring," which is a pair of groups where the function of one group distributes over the function of the other group. All the theorems of group theory can be applied to logic using Boolean algebra. Boolean algebra is also the basic mathematics that is used inside computers to describe how arithmetic is done and how hardware is designed.