In the center of the Earth, there are equal amounts of mass in all directions at any given distance, so gravity pulls equally hard in all directions, so an object at the center of the Earth would feel zero net gravity. Such an object would be under enormous pressure, because it does feel the weight of all mass that lies on top of it.
Outside of the Earth, gravity behaves as if all mass of the Earth were concentrated in the very center point, but inside the Earth gravity does not behave like that.
Gravity comes from mass, not from a particular point.
Only in the precise center is the gravity equal to zero.
This holds also in the center of a
black hole, if the law of gravity as we know it is still valid there (which we don't know).
If the Earth were hollow inside, and all of its mass were in the surface shell, and the mass distribution was spherically symmetric, then you'd be
weightless everywhere in the hollow space inside the Earth.
The green line in
http://www.splung.com/kinematics/images/gravitation/variation%20of%20g.png shows the variation of the strength of gravity with distance from the center of the Earth.
Gravity is zero in the very center, then rises fairly linearly to about 109% of the surface gravity at about 55% of the distance to the surface, then drops back to about 100% and stays there until the surface. Beyond the surface the gravity decreases with the square of the distance from the center.
The red line in the same graph shows the variation of gravity if the Earth were equally dense everywhere. Then the strength of gravity would increase linearly from 0 in the middle to 100% at the surface. That in the "real" Earth the variation of gravity is different is because the Earth is not equally dense everywhere, but is a lot denser near the middle than near the surface.