Your weight is the normal force the ground pushes up on you to keep you falling into the center of the earth. A person on the equator is in circular motion. They must have some force (gravity) pulling them into the center to keep them in circular motion. It turns out gravity is a little too strong, so the ground supplies some normal force to keep them from spiralling in. This normal force, less than the full force of gravity, is the weight.
As a silly example, if the earth spun at a million miles an hour, people on the equator would be flung into outer space and be weightless.
I hope this helps. I answered this question last year and some people still didn't get it.