In thermodynamics, we find all kinds of important named concepts and equations that go back to this great American scientist of the nineteenth century (some people even go so far as to say he is the greatest American scientist, ever).
The Wikipedia entry does a competent job of recounting the man’s scientific achievements — he singlehandedly created the field of chemical thermodynamics. Also he had a big hand in the early development of vector algebra and analysis, and statistical mechanics. [BTW, this entry at the Encyclopedia of Human Thermodynamics is pretty good too. It has a short section devoted to anecdotes about JWG.]
He was born in a family with a long line of scholars. His father was a renowned linguist at Yale (he played a role in the Amistad case whose origins date back to the year JWG was born). JWG grew up in New Haven, studied at Yale for his bachelors and doctoral degrees. He joined Yale as a tutor, and except for 3+ years of European visit during 186669, he rarely ventured out of New Haven. He remained unmarried.
I was surprised to learn that his early years as a faculty member at Yale were primarily in an honorary (i.e., unpaid) capacity. Yale made him a (properly salaried) professor only when he received a professorship offer from Johns Hopkins! JWG was into his middle years by then (1880, when he was 41), and all his three major pieces on thermodynamics had been published.
Since those papers appeared in the Transactions of the Connecticut Academy, they never reached the kind of scientific audience they deserved; it took the efforts of some of his cheerleaders — including James Clerk Maxwell! — to make them well known among scientists in Europe.
Here are a few excerpts on the man and his science:

… He had studied with Weierstrass and was not unmindful of mathematical rigor; in the paper in which he pointed out that phenomenon of the convergence of Fourier series which has come to be known as the Gibbs phenomenon he showed his appreciation of mathematical precision as he did on other occasions. But fundamentally he was not interested in rigor for itself, he was inspired by the greater problem of the union between reflective analytical thought and the world of fact. He did not feel that one should not study pure mathematics; he was not onesided or dogmatic in any of his views. What he said was that one of the uses of a good mathematical background was in the study of the problems set us by Nature. [E.B. Wilson, Ref. 1, below]

“A mathematician may say anything he pleases, but a physicist must be at least partially sane.” This quote, attributed to JWG (see Wikiquotes), goes quite well with Wilson’s recollection of JWG’s attitude towards math and physics.
References:
 Edwin Bidwell Wilson: Reminiscences of Gibbs by a student and colleague
 Biographical Memoir of Josiah Willard Gibbs, published in the Biographical Memoirs – National Academy of Sciences, Volume VI (1909).
 The Greatest Simplicity: J. Willard Gibbs in Great Physicists: The Life and Times of Leading Physicists from Galileo to Hawking by William H. Cropper, Oxford University Press (2001).