Hermann Günther Grassmann was born in 1809 in Stettin, near the border of Germany and Poland. He was only 23 when he discovered the method of adding and multiplying points and vectors which was to become the foundation of his Ausdehnungslehre (extension theory). In 1839 he composed a work on the study of tides entitled Theorie der Ebbe und Flut, which was the first work ever to use vectorial methods. In 1844 Grassmann published his first Ausdehnungslehre (Die lineale Ausdehnungslehre ein neuer Zweig der Mathematik) and in the same year won a prize for an essay which expounded a system satisfying an earlier search by Leibniz for an 'algebra of geometry'. Despite these achievements, Grassmann received virtually no recognition. This was no doubt in some part because his ideas and notation were too new.
In 1862 Grassmann re-expounded his ideas from a different viewpoint in a second Ausdehnungslehre (Die Ausdehnungslehre. Vollständig und in strenger Form). Again the work was met with resounding silence from the mathematical community, and it was not until the latter part of his life that he received any significant recognition from his contemporaries. Of these, most significant were J. Willard Gibbs who discovered his works in 1877 (the year of Grassmann's death), and William Kingdon Clifford who discovered them in depth about the same time. Both became quite enthusiastic about this new mathematics.
An account of Grassmann's life woven with the lives of other mathematicians instrumental in the development of modern vector analysis is given by Michael Crowe:
A brief biography of Grassmann is contained in the book Grassmann Algebra Volume 1 and reproduced on the Biography page of grassmannalgebra.com. This page also contains the references to the comprehensive biographies of Grassmann by Hans-Joachim Petsche.
A conference was held to celebrate the 200th anniversary of Grassmann's birth in Potsdam, Germany in September 2009. Here is a download link to a short talk I gave: Grassmann, Geometry and Mechanics.pdf