When you're at the surface, all of the Earth is below you and pulls you down. At the center you're surrounded in all directions, the pull in all directions would be equal, and you would experience no net gravity.
As you descend from the surface to the center the gravity would gradually decrease. Only the part under you attracts you. E.g., when you're at the surface, you're on a sphere with a 4000 mile radius (rounded), and all of it attracts you. If you're 1000 miles down, the outer 1000 miles of the Earth has no net effect (it's pulling in all directions equally), and only the 3000 mile radius sphere under you is attracting you. As you get closer to the center, more of the Earth is above you and doesn't affect you until at the center, you feel no gravity.
You'd probably feel a great deal of heat and pressure, though. At least for a fraction of a second.
Add:
Rochester is correct about still feeling the Moon's gravitational attraction. It would exert a force of around 7 thousandths of one ounce. You'd also feel the gravity of the Sun (much larger - around an ounce), plus minor contributions from Jupiter, the other planets, and every other piece of matter in the universe. He's wrong about where you'd have to go to feel no net effect, though (e.g., the center of the earth-sun system is inside the Sun, but you can feel no net gravity from that system very near the center of the earth by moving a tiny bit toward the Sun).
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