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Descartes' biggest contribution to math is analytic geometry. He figured out how to plot points on a plane, what we now call the Cartesian coordinate plane. Descartes' contributions are what led to the revelations of calculus, 50 years after his book, A Discourse on the Method of rightly conduction the Reason and seeking Truth in the Sciences, was published.
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Pascal is best known for Pascal's triangle, and before he was even 16 years old he had proved a theorem: "It states that given a (not necessarily regular, or even convex) hexagon inscribed in a conic section, the three pairs of the continuations of opposite sides meet on a straight line, called the Pascal line" (Smith, History of Mathematics). Pascal also helped lay the foundation for the theory of probability. (The quote used was taken directly from Professor Bedard's summary.)
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Tschirnhaus worked in analytic geometry and solving polynomial equations. He found a method of solving cubic equations using a Tschirnhaus transformation. It is also thought that Tschirnhaus invented European porcelain.
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Maclaurin worked in the fields of calculus, geometry, and gravitation, expanding upon Newton's own work. He also "greatly generalized the theory of the mystic hexagram" (D.E. Smith, History of Mathematics).
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Euler was the first to introduce the concept of a function, though not as it is thought of today. He also defined polynomial, trigonometric, exponential, and logarithmic functions.
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Being the first woman to publish a book about math, Maria also was one of the first to discuss and summarize calculus up in her time. One of her best known contributions to math is a plan curve called the Witch of Agnesi, though she herself didn't discover it, merely mentioning it in her book.
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Lagrange published "Mécanique Analytique" and also contributed to the metric system and his introduction of the decimal base; some consider him the father of the metric system. He also worked on planetary motion and on ways to rewrite Newton's Equations of Motion.
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Sophie Germain is noted for her work in acoustics, elasticity, and the theory of numbers. Sophie Germain primes are named after her. In 1819 she wrote detailing her strategy for solving Fermat's last theorem.
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Cauchy came up with the concept of the limit, which was discovered long after the actual invention of calculus. His definition of a limit was: "When the values successively attributed to the same variable approach indefinitely a fixed value, eventually differing from it by as little as one could wish, that fixed value is called the limit of all the others." Cauchy also is attributed with being the first to use imaginary numbers.
