The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. 

The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. 

It was founded in 1888 as the "New York Mathematical Society", the brainchild of Thomas Fiske, who was impressed by the "London Mathematical Society" on a visit to England. John Howard Van Amringe was the first president and Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance, due to concerns about competing with the American Journal of Mathematics. The result was the "Bulletin of the American Mathematical Society", with Fiske as editor-in-chief. The de facto journal, as intended, was influential in increasing membership. The popularity of the "Bulletin" soon led to Transactions of the American Mathematical Society and Proceedings of the American Mathematical Society, which were also "de facto" journals. 

In 1891 Charlotte Scott became the first woman to join the society. The society reorganized under its present name and became a national society in 1894, and that year Scott served as the first woman on the first Council of the American Mathematical Society.
And what did they become at that time?
American Mathematical Society