To answer your questions properly:
Q: What is gravity?
A: The natural force of attraction exerted by a celestial body, such as Earth, upon objects at or near its surface, tending to draw them toward the center of the body.
Q: Where is it from (Source)?
A: It comes from the center of the body, where the mass pulls in objects.
Q: Why does gravity push up down not up?
A: Gravity does not push up down, but only pulls. Down can only be used because there is a sense of direction from a point that is not space, hence, in this case, up into the air and back to the ground.
Q: Why is there no gravity in space?
A: Space is what it literally means, space. There is nothing at all, except to hold matter, antimatter, energy, and dark energy.