Math Resources

Wolfram MathWorld  - math encyclopedia - for advanced students 
Art of Problem Solving (AoPS)  - Forums on a wide range of mathematical topics and levels; very large database of problems, including Olympiad problems; listings of books, competitions, summer programs, scholarships etc.; math and problem-solving classes; AoPS Alcumus series of lectures and problems; and the AoPS Wiki, a useful non-technical math encyclopedia.  All free -except some of the classes. 
The Math Forum  – wide range of math resources.
Planet Math  - online encyclopedia.
Cut the Knot - math-related articles, puzzles, games, and demonstrations.
The Dynamical Systems and Technology Project  - Boston University resource for students and math teachers.
Pi in the Sky – math magazine for high schoolers.
Daily online SET game

National or International Math Competitions
American Mathematics Competitions - AMC 8, AMC 10, AMC 12, AIME, USAMO (proof-based), and IMO (International Mathematical Olympiad).  This set of competitions is for students from below 8th grade to 12th grade.
American Regions Mathematics League (ARML)
USA Mathematical Talent Search (USAMTS) – proof based, free, online, and untimed
Mandelbrot Competition 
Mathcounts - for middle school students
MathLinks Extremely challenging, free, proof-based, online, AoPS-run, weekly contest 
Siemens STS Competition  – research competition
Intel Science Talent Search– research competition

            Regional Math Competitions
Math League - various state and regional math competitions at elementary, middle, and high school levels
Bay Area Math Olympiad (BAMO)
Colorado Mathematical Olympiad 
Harvard MIT Math Tournament (HMMT)
Mu Alpha Theta - regional, state, and national contests
Princeton Math Competition (PUMaC)
Rice Math Tournament

Other Math Summer Programs 
The Ross Program
Hampshire College Summer Studies in Mathematics (HCSSiM)
Research Science Institute 
Minority Introduction to Engineering and Science (MITES)
Canada/USA Mathcamp
 AwesomeMath
MathZoom Summer Camp 
Math Path - for middle school students.
SEARCH at Mount Holyoke College - summer program for girls 
Stanford University Mathematics Camp (SUMaC) 
American Mathematical Society's listing of high school summer programs