Frege is arguably the greatest logician since Aristotle.
His revolutionary Begriffsschrift (Concept Script) from 1879 marked the beginning of a new epoch in the history of logic by displacing the old Term Logic that had held sway virtually unchanged since Aristotle. The Begriffsschrift was ground-breaking, and made contributions that are nowadays ubiquitous in mathematics, such as the use of quantification and a clean treatment of functions and variables.
However, Frege's two-dimensional notation was so idiosyncratic that nobody uses it today.
Nonetheless, some vestige of his notation survives: the symbol that logicians informally call
"turnstyle" is derived from Frege's "Inhaltsstrich".
Frege was the first to devise an axiomatization of propositional logic and of predicate logic, the latter of which was his own invention. The quantification so essential to Bertrand Russell's theory of descriptions, and to Russell and Alfred North Whitehead's Principia Mathematica, was also due to Frege. His work was largely unrecognized in his own day, and his ideas spread chiefly through those he influenced, particularly Giuseppe Peano and Russell. Ludwig Wittgenstein and Edmund Husserl were among the other philosophical notables strongly influenced by Frege.
Frege was the first major proponent of logicism -- the view that mathematics is reducible to logic. His Grundgesetze der Arithmetik was an attempt to explicitly derive the laws of arithmetic from logic. After the first volume was published (at the author's expense), Russell discovered the paradox which bears his name, and that the axioms of the Grundgesetze led to this contradiction; he wrote to Frege, who acknowledged the contradiction in an appendix to volume two of the Grundgesetze, noting what he perceived to be the faulty axiom. Frege never did manage to amend his axioms to his satisfaction, however; and after Frege's death, Kurt Gödel's incompleteness theorems showed that Frege's logicist program was impossible.
Frege was born in Wismar. He started studying at the University of Jena in 1869 and moved to Göttingen after two years, where he received his Ph.D. in 1873. After returning to Jena two years later, he became lecturer of mathematics. In 1879, he was made associate professor and in 1896 became professor of mathematics. He died in Bad Kleinen in 1925.
His principal works are:
Begriffsschrift (Concept Script), eine der arithmetischen nachgebildete Formelsprache des reinen Denkens, Halle a. S., 1879
Die Grundlagen der Arithmetik (The Foundations of Arithmetic): eine logisch-mathematische Untersuchung über den Begriff der Zahl, Breslau, 1884
"Funktion und Begriff" ("Function and Concept"): Talk given in a Meeting on January 9, 1891 of the Jenaischen Gesellschaft für Medizin und Naturwissenschaft, Jena, 1891
"Über Sinn und Bedeutung" ("On Sense and Denotation"), in Zeitschrift für Philosophie und philosophische Kritik, C (1892): 25-50
"Über Begriff und Gegenstand" ("On Concept and Object"), in Vierteljahresschrift für wissenschaftliche Philosophie, XVI (1892): 192-205
Grundgesetze der Arithmetik ("Basic Laws of Arithmetic"), Jena: Verlag Hermann Pohle, Band I (1893), Band II (1903)
Was ist eine Funktion? ("What is a Function?"), in Festschrift Ludwig Boltzmann gewidmet zum sechzigsten Geburtstage, February 201904, S. Meyer (ed.), Leipzig, 1904, pp. 656-666
"Der Gedanke" ("The Thought") Eine logische Untersuchung, in Beiträge zur Philosophie des Deutschen Idealismus'' I (1918-1919): 58-77
"Die Verneinung" ("Negation"), in Beiträge zur Philosophie des deutschen Idealismus I (1918-1919): 143-157
"Gedankengefüge" ("Compound Thought"), in Beiträge zur Philosophie des Deutschen Idealismus III (1923): 36-51
Frege intended these last three papers to be published in a book to be called Logical Investigations; in 1975 they were posthumously published (in English translation, at least) under this title.