The Newtonian view of space is a rigid three dimensional structure, which can be expressed as three perpendicular directions. Time is separate from the space dimensions. Time permeates all of space, moving at the same rate everywhere. Gravity in this structure appears as a force, acting between any two masses as an inverse square law. This force causes objects to accelerate.
General Relativity tells us that spacetime is a unified four dimensional structure. The presence of matter causes spacetime to become warped. Bodies moving in this warped spacetime appear to be subject to a force because they do not travel in straight lines, but they are actually moving along special curves in spacetime called geodesics. They appear to be subject to a force, but in fact they are moving in warped spacetime. Picture for example, the warping of spacetime by a large mass – say a planet, as below right (the image shows warping in two dimensions, though of course, spacetime is actually four dimensional).
As a particle moves in the region of the mass, it will follow a curved path, being deflected towards the mass/ If the mass increases, the curvature of spacetime becomes more pronounced, and the amount of deflection increases. The warping on a two dimensional sheet can be used to illustrate the orbit of the Earth around the Sun.
The Sun warps space time and the Earth follows a geodesic, a path of shortest possible length in four dimensional spacetime, which in this case is a closed path.