Too many to count (har har).

This is an impossible question to answer since there is no universal objectively measurable definition of 'genius.' It's also hard to say if anybody 'changed' the course of history since it's entirely possible that had one mathematician failed, others would have made similar deductions and the overall path of technological development would not have been drastically altered.

Now if we just restrict our question a little and inquire instead about mathematicians whose work made a significant impact on human civilization, we can start to get somewhere. But that list is long indeed. I'll just list some off the top of my head in a somewhat haphazard way, while conceding that this is a very partial list at best, and with the hope that others can add to it.

Euclid wrote Elements and made early contributions to geometry. Euclidean geometry, even today, refers to the normal "flat" space that uses the Pythagorean theorem (and its N dimensional equivalents) as its metric (i.e. distance between points). The study of euclidean geometry is of obvious importance since it is the most straightforward way to represent our world in the way we are familiar with. Elements has been used as a textbook in geometry until fairly recently.

Newton and Leibniz independently derived what we now call calculus. Its immediate application was in the description of Newtonian mechanics (or essentially the physics of motion at 'low' or non-relativistic speeds). Our modern day notations derive from the work of Leibniz. Calculus in turn leads directly into concepts of continuity from which we develop more robust concepts that fall under the field known as analysis. Anlysis would go on to be formalized and developed by figures like Dirichlet, who also studied practical solutions to differential equations (any equation involving rates of change). Newton is also the inventor of Newton's method, an early numerical algorithm for computing the root of continuous functions (i.e. it provides a way to solve equations too hard or impossible to do analytically). Calculus, analysis, and numerical methods are ubiqutious and foundational concepts in every field of science and engineering.

Some early figures in the study of probability and modern combinatorics include Pascal, Fermat, Bernoulli and de Moivre. The importance of probability probably doesn't need to be emphasized for anybody who reads the news. It forms the basis of statistics as well as stochastic algorithms (i.e. computing the output of difficult problems using random guesses). Fermat went on to do important work on numerical integration, and number theory (the field which, most notably, underlies modern cryptography). Bernoulli also did work on differential equations and infinite series (sums of infinitely long sequences -- which may be used to model a great deal of natural phenomena) as well as discovering the natural logarithm e which is ubiquitous in mathematical identities across many fields (although it should be noted it would be the work of others that made those connections).

Linear algebra is likewise a foundational field of modern science and engineering, along with analysis and probability. Systems of linear equations have been around since ancient times. Leibniz discussed earlier first introduced the modern definition of determinants which are single numbers that describe a great many properties of linear systems / transformations / matrices (these are all equivalent). Gauss to be discussed below formalized gaussian elimination which as it turns out, is the first known matrix decomposition technique, or the LU decomposition (for lower / upper triangular matrices). Matrix decompositions are the basis of modern computational solutions to linear algebra problems (and linear problems can be used to approximate many non-linear problems). The concept of a vector space was introduced by Peano which unified many elements of linear algebra and now forms the foundation for students of the field.

Algebra in theoretical math refers to the abstract study of operations between elements of generalized sets. As it turns out a great many different types of algebras share the same rules -- matrices, functions, real numbers, complex numbers, differential operators, and so on -- can all be unified with the study of algebra leading to insights and advances not otherwise obvious. Euler and Gauss to be discussed below both studied non-traditional algebras (modular arithmetic in the context of number theory). Kronecker formalized many core concepts that now underlie the field.

The great polymath Euler contributed so much to mathematics that he gets his own paragraph. Wikipedia has a list of things named after Euler to give you some idea of just how prolific he was. One thing that stands out is his contribution to complex analysis as he proved some key identities involving the imaginary number i. Complex analysis is of crucial importance to many fields of science and engineering. Most familiar to the average person is probably the field of signal processing, as all waves including electrical waves, light waves, and sound waves can be represented with complex algebra. This is typically done through fourier analysis named after Joseph Fourier, which transforms between spatial representations of waves and frequency-phase representations of waves. Euler also made key advances in real analysis, geometry, and number theory among other things.

Our other great polymath is Gauss who also has an impressive list on Wikipedia. Gauss was an early inventor of the fast fourier transform which enables the aforementioned fourier transform to be computed on modern computers in a reasonable amount of time. As the fast fourier transform is discrete, it predates the fourier transform itself (which operates on continuous functions). Gauss also did work on non-euclidean geometry, where normal metrics and axioms of euclidean geometry does not apply. This opens up the rigorous study of projective geometry and generalized metric spaces and through it the mathematical formulation of relativity. As an aside, modern computer graphics and computer vision (i.e. your phone, camera, movies and video games) rely on projective geometry as represented by homogeneous coordinates introduced by Möbius who is more famous for topology. The gaussian distribution is ubiquitous in science as it not only models a great deal of natural phenomena but has some very beautiful properties (such as being transformed into another gaussian distribution under fourier transform) that give it enormous utility.

Set theory is the basis of modern mathematics. Many earlier mathematicians, including ones discussed here, used proofs with a wide range of axioms and less rigorous standards of proof than would be acceptable today. Modern mathematics on the other hand can be developed from a small set of axioms which define sets. Cantor is a key figure in the development of set theory. Cantor demonstrated the abstract existence of multiple types and rankable 'sizes' of infinity, which sparked something of an existential crisis among serious mathematicians. Other key figures in the development of foundational mathematics include Russell and Gödel.

Some other crucial contributions deserve mention. Modern numerical analysis owes much to Taylor approximation which owes its development to Taylor, Lagrange and Cauchy. Taylor approximation is a method for approximating any continuous function in a local neighborhood using a series of polynomials. This leads to rigorous bounds on the error of various numerical integration schemes, without which modern engineering can not exist. Cauchy also was an originator of modern complex analysis which was discussed earlier. Lagrange is an early investigator of variational calculus which seeks to generalize calculus not just to functions of one or a few variables, but infinitely many variables, i.e. functions of continuous functions. Variational refers to the finding of minima and maxima -- which in the form of optimization is enormously useful to modern science. Jacobi applied calculus to the study of functions of multiple inputs and outputs (and hence differential geometry) through the jacobian. Poisson did extremely important work in statistics / probability, waves, and analysis which was (and is) immediately applicable to physics. Laplace likewise made breakthroughs in linear algebra, analysis, and probability. Of direct application, he demonstrated analytical solutions of many integrals. His work on least squares problems became a foundation of modern science as soon as computers became a thing. His work on conditional probability is directly applicable to spam filtering, among other things.

As we move into the 20th century mechanical, analog, and then digital computers spread around the world, which is a great boon for math and its applications. I'm going to run out of space here but if there's interest I might do another post focusing on the mathematics and mathematicians of the modern era, particular in applied fields and computer sciences, as they have had a profound impact on human civilization as it exists today.

/u/flyingdragon8 has given you a good list of names, so let me make a remark concerning your first question. That is, whether asking for an "all-star list" is reasonable in the first place. Just as for history in general, there are a number of "great men" narratives in mathematics that are popular even among professionals. Temple Bell's once popular Men of Mathematics comes to mind as an oft-criticized piece of this sort of "history" of mathematics. In general, closer examination often reveals that groundbreaking or surprising results in mathematics are the consequence of long lines of research and ideas. How these ideas develop over time is sometimes not even obvious to the people directly involved.

Let me give the examples of two very successful theories that you might have heard of: Einstein's special relativity (the article is a Poincaré seminar article by Darrigol) and Fisher's maximum likelihood, (an article in Stigler in Statistical Science). In both cases, even specialized textbooks rarely acknowledge or mention anyone other than Einstein or Fisher. Heated priority disputes are very frequent even in pure math (for another semi-famous example you can look at the Erdos-Selberg controversy in the late 1950s. More often than not, after a few years, only a few names out of many contributors remain attached to any result. Misnamed theorems (article by Katz about the famous integral theorems of Gauss, Green and Stokes) are very frequent.

Sorry, we don't allow throughout history questions. These tend to produce threads which are collections of trivia, not the in-depth discussions about a particular topic we're looking for. If you have a specific question about a historical event or period or person, please feel free to re-compose your question and submit it again.