Google Scholar is a search engine for academic publications and also provides citation counts.
AMS MathSciNet is an online bibliographic reference for mathematical journal articles and publications. (Subscription required)
Mathematics ArXiv is an online repository of mathematical preprints (articles that are not yet published in a journal or book).
The Mathematics Genealogy Project shows the mathematical ancestors and descendants of a mathematician.
Institute for Advanced Study is one of the world's leading centers for theoretical research, especially pure mathematics.
Mathematical Sciences Research Institute is an NSF-funded institute located in Berkeley, California.
Clay Mathematics Institute is a privately funded foundation dedicated to supporting mathematical research and disseminating mathematics. It is best known for posing the Millennium Problems setting a $1 million prize for the first solution to each problem.
American Institute of Mathematics is a privately funded research institute in Palo Alto, California.
Institut des Hautes Etudes Scientifiques located in Bures-sur-Yvette, France (just outside Paris) is one of the world's leading mathematical research institutes.