Here is a partial list of math competitions:

MATHCOUNTS, run by the Mathcounts Foundation

Math Kangaroo

Mathleague.com

Mathleague.org

AMC 8,

AMC 10

AMC 12

Mandelbrot- no longer running

iTest-No longer running

Online Math Open-No longer running

AIME

USAMO

USAMTS, run by the NSF (with a little help from Art of Problem Solving)

ARML (Also ARML Power Contest and ARML Local)

MOEMS

Rocket City Math League

OMMC- Online Monmouth Math Competitions

Purple Comet

Mu Alpha Theta

American Masters in Math

Math Prize for Girls

Interschool Tests

School:
Hoover High Math Competitions
Vestavia Hills Math competitions
Grissom High Math Competition
UAB Math Talent Search
MIT Integration Bee
Harvard MIT Math Tournament (November and February)

State:
Alabama Statewide High School Math Contest
FAMAT

School:
Exeter Math Club Competition

China:

World Math Team

Undergraduate:
Putnam

IMC (International)