Below I list some websites that I visit very often. They are not very neatly categorized. I may do so at some point in time.
The Abdus Salam International Centre for Theoretical Physics
Chennai Mathematical Institute
Harish-Chandra Research Institute
The Institute of Mathematical Sciences
The value of the variable: H. A. Helfgott
British Combinatorial Committee
The Electronic Journal of Combinatorics
The Mathematical Intelligencer
European Journal of Combinatorics
The Journal of Irreproducible Results