Math Links

Art of Problem Solving (AoPS) - forums, problems, lectures, information and courses on a wide range of topics and levels.
Math StackExchange - Q & A forum for a wide range of mathematics
Wolfram MathWorld - math encyclopedia - for advanced students
Khan Academy - thousands of short instructional videos 
The Math Forum  - wide range of math resources 
Curriculum Inspirations - multimedia experience from MAA for middle-school and high-school communities
National Math Club - provides resources and materials to run a math club
Planet Math  - online encyclopedia
Plus Magazine published by Cambridge University
Association for Women in Mathematics
NRICH - online mathematical resources and forums in the UK.
ArXiv - open access to over 800,000 e-prints in math and math-related fields.

Some Math Competitions

American Mathematics Competitions - AMC 8, AMC 10, AMC 12, AIME, USAJMO, USAMO, and IMO (International Mathematical Olympiad).  This set of competitions is for students through 12th grade.
American Regions Mathematics League (ARML)
USA Mathematical Talent Search (USAMTS) – proof based, online, untimed.
Math Prize for Girls - for high-school girls 
Harvard MIT Math Tournament (HMMT) 
Princeton Math Competition (PUMaC)
Mandelbrot Competition
Math League
 - runs various state and regional math competitions at elementary, middle, and high school levels.
Mathcounts - for middle school students
Math Video Challenge - team competition for middle school students
Bay Area Math Olympiad (BAMO)
Mu Alpha Theta - runs regional, state, and national math contests.
Regeneron Science Talent Search – research competition (used to be Intel STS)
Purple Comet! Math Meet - online, international, team competition
European Girls Mathematical Olympiad (EGMO) - international competitiona
China Girls Mathematical Olympiad (CGMO) - international competition
International Tournament of Young Mathematicians - international open problem team competition
AoCMM - Online team contest of mathematical modeling

Other Math Summer Programs 

All Girls, All Math
AwesomeMath
Canada/USA Mathcamp
Camp Euclid - online research program for middle and high school students
Hampshire College Summer Studies in Mathematics (HCSSiM)
MathILy
MathPath - for middle school students
Mathworks Honors Summer Math Camp
MathZoom Summer Camp
Minority Introduction to Engineering and Science (MITES)
Research Science Institute (RSI)
Ross Mathematics Program
Simons Summer Research Program - STEM research program for rising seniors
Stanford University Mathematics Camp (SUMaC)
Williams College Math Camp (WCMC)

The American Mathematical Society's listing of high school summer programs is available here.  

Some School-Year Math Options

PRIMES - year-long mentored research program. 
MIT Open Courseware - MIT course materials online.
Udacity - online courses in math and science.
Coursera
 - online courses.

Math Circles - click here to look for a math circle near you.