The Berkeley Math Circle

Links

Summer Programs

SPARC 
A* Summer Math Camp (Orange County and Bay Area)
A* Summer Computer Science Camp (Bay Area)
Hampshire College Summer Studies in Mathematics (Amherst, MA)

Elementary School: 

Epsilon Camp (Colorado)

Middle School:

Math Path

High School:

Bay Area or nearby

AwesomeMath (UC Santa Cruz and Cornell University) 
COSMOS (UC Davis) 
SUMaC (Stanford)
Math Zoom Summer Camp (Cal Poly Pomona and Rutgers)

Abroad

Modern Mathematics - International Summer School for Students (Jacobs University, Bremen, Germany)

Elsewhere

Candada/USA Math Camp (Mount Holyoke College in South Hadley, MA) 
Bright Math Camp
Summer Program for Women in Mathematics (Princeton) 
Hampshire College Summer Studies in Mathematics
MOSP USA team preparation for the IMO
PROMYS (Boston University)
The Ross Program
Young Scholars Program
Research Science Institute (most advanced in research)
All Girls/All Math
International Math Summer School (Germany) (brochure)

Math Competitions:

American Math Competitions
AIME
ARML
Canadian Math Competition
International Math Olympiad
Mrandelbrot Competition
MATHCOUNTS
Putnam Competition
USA Math Talent Search
USAMO
Math Kangaroo
ABACUS
MOEMS
North American Computational Linguistics Olympiad
The Stanford Math Tournament
ProCo - High School Programming Contest at Stanford

Institutes & Organizations:

Mathematical Sciences Research Institute
Pacific Institute for Mathematical Sciences
American Institute of Mathematics
The Clay Math Institute
American Mathematical Society
Mathematical Association of America

Math Journal:

Mathematical Reflections
Notices of the American Mathematical Society
American Mathematical Monthly

Conferences:

MSRI National Conference
AMS/MAA Conference in New Orleans 

Online Math Resources
Art of Problem Solving
MathLinks
IMOmath