Math Links

The Math Forum  - wide range of math resources 
Planet Math  - online encyclopedia
Art of Problem Solving (AoPS) - forums, problems, lectures, information and courses on a wide range of topics and levels. 
Wolfram MathWorld - math encyclopedia - for advanced students
Khan Academy - thousands of short instructional videos 
Plus Magazine published by Cambridge University
Math StackExchange - Q & A forum for a wide range of mathematics
Association for Women in Mathematics
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 and HMNT) 
Princeton Math Competition (PUMaC)
Mandelbrot Competition
Math League
 - runs various state and regional math competitions at elementary, middle, and high school levels.
Bay Area Math Olympiad (BAMO)
Mu Alpha Theta - runs regional, state, and national math contests.
Siemens STS Competition  – research competition. 
Intel Science Talent Search – research competition.
Purple Comet! Math Meet - online, international, team competition
European Girls Mathematical Olympiad (EGMO) - international competition
China Girls Mathematical Olympiad (CGMO) - international competition
International Tournament of Young Mathematicians - international open problem team competition

Other Math Summer Programs 

Ross Mathematics Program
Hampshire College Summer Studies in Mathematics (HCSSiM)
Research Science Institute (RSI)
Minority Introduction to Engineering and Science (MITES)
Canada/USA Mathcamp
Stanford University Mathematics Camp (SUMaC)
Honors Summer Math Camp
AwesomeMath
MathZoom Summer Camp 
Math Path - for middle school students.
Camp Euclid - online research program for middle and high school students.
Modern Mathematics International Summer School for Students (MOMISSS) 

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. 

A Few Year-Round Math Programs and Resources in Europe

Animath - online French mathematics resources and contests.
NRICH - online mathematical resources and forums in the UK.
KöMaL - Exercises and problems in Mathematics, Informatics and Physics for advanced high school students.
Estalmat - program for mathematically gifted students in Spain.
Berlin Network of Schools Specializing in Mathematics and Science
Germany Mathematical Society (DMV)
Math en Jeans - team research program in France.