The gravitational force inside a uniform shell of mass (due to the shell itself) is zero at all points inside the shell. The gravitational force due to each point on the sphere balances out just right so that the net force is zero. This means that the gravitational force at the exact center of a spherical body like the Earth is zero. It also means that if you dig part way down, gravity gets weaker since you only feel the force of the mass at a smaller radius than where you're standing.
http://www.reddit.com/r/askscience/comments/nd4lw/if_it_was_possible_to_stand_in_the_center_of_the/
http://www.reddit.com/r/askscience/comments/n2gj9/assuming_it_was_possible_how_would_gravity_effect/
http://www.reddit.com/r/askscience/comments/mraqc/how_does_gravity_get_strongerweaker_as_you_get/
http://www.reddit.com/r/askscience/comments/lo7ix/gravity_if_the_center_core_of_the_earth_was/
http://www.reddit.com/r/askscience/comments/kt9am/in_theroy_what_would_happen_if_you_were_at_the/
http://www.reddit.com/r/askscience/comments/k95p8/what_would_gravity_do_to_you_at_the_center_of_the/
http://www.reddit.com/r/askscience/comments/jj58x/what_would_happen_to_gravity_at_the_center_of_the/
http://www.reddit.com/r/askscience/comments/hsrsq/what_would_happen_in_terms_of_gravity_if_you/
http://www.reddit.com/r/askscience/comments/hbgwr/is_the_pull_of_gravity_weaker_near_the_center_of/
http://www.reddit.com/r/askscience/comments/gcyyo/theoretically_if_you_could_go_down_to_the_very/
http://www.reddit.com/r/askscience/comments/fn048/if_you_drilled_a_hole_halfway_to_the_center_of/
http://www.reddit.com/r/askscience/comments/ea9w7/question_about_gravity_as_you_get_close_to_the/
http://www.reddit.com/r/askscience/comments/cmu5s/at_the_center_of_the_earth_do_you_experience_zero/
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