Spacetime is the 4-dimensional vector space or manifold that simultaneously embodies the concepts of time and 3-dimensional space. Whether one can ascribe a separate existence to spacetime itself, depends on one's physical intuition. But that doesn't matter—"separate existence" or not, physicists working in the field understand spacetime as a useful concept.
Under Newtonian/Galilean physics or Newtonian/Galilean relativity at the high-school and general undergraduate level, one typically thinks of time, and the 3 dimensions of space, completely separately. One can solve many problems of physics with this formulation.
But a more sophisticated view of "spacetime" as a single entity allows one to give a very elegant formulation of fictitious forces. A "fictitious force" is a perceived force on a body in the absence of any visible agent pushing on it. Examples are the centrifugal force and the Coriolis force. These forces all have the property that they are exactly proportional to the object's mass. With careful analysis under Newtonian/Galilean physics, it's not hard to see why. The "force" arises from the acceleration of the frame of reference. Gravity also satisfies the criterion of being exactly proportional to mass—this is Einstein's "equivalence principle".
When the physical system is formulated as "spacetime", that is, something with spatial and time coordinates, the acceleration can be seen to be just an aspect of the curvature of the coordinate system. In cases like the simple centrifugal or Coriolis force, the curvature of the coordinate system can be "transformed away" by analyzing the situation in a non-accelerating frame of reference. For example, if one stands outside of an amusement park ride, one can easily see the coordinate curvature effects that cause people on the ride to perceive a force. (This has nothing to do with Einsteinian relativity. It arises in pure Newtonian/Galilean physics.)
This "transforming away" is similar to the difference between Cartesian and polar coordinates on a flat sheet of paper. The Cartesian coordinate system is flat, and the polar system is curved. But they are both coordinate systems on a flat sheet of paper. However, there are surfaces ("curved surfaces"), like a sphere, that have no flat coordinate system at all. In spacetime, this is equivalent to something that has no flat coordinate systems, and hence gives rise to fictitious forces that can't be transformed away. Gravity is an example of this, a fictitious force that, unlike the centrifugal or Coriolis force, can't be transformed away. That is, there is no place where one can stand and see gravity as just a case of people being pushed on by a floor that is accelerating upward.
The curvature of spacetime refers to this. It gives rise to the "fictitious forces" that are perceived as gravity under General Relativity. The Einstein Field Equations explain how that curvature arises from the presence of massive objects like the Earth or the Sun. Analyzing the resultant gravitational fictitious force arising from massive objects obtains the same result as Newton's law of gravity for non-relativistic objects.
- Often treated as a vector space when working examples, but that isn't really correct because it has no origin. The more sophisticated term "manifold" is correct.
- The bane of mapmakers.