A central force is simply a force that is always directed towards a fixed point in space. Gravity can be treated as a central force in certain circumstances.
The gravitational force acting on object $A$ orbiting around object $B$ can be approximated by a central force acting towards the centre of $B$ if $B$ is much more massive than $A$. By Newton's third law the gravitational attraction of $A$ on $B$ is equal and opposite to that of $B$ on $A$, but if $B$ is much more massive than $A$ then to a first approximation we can assume that $B$ is stationary.
It is also true that in the two-body problem, where $A$ and $B$ have similar masses, then they will orbit around a fixed point or barycenter. The gravitational attraction of one object on the other still acts as a central force directed towards this barycenter, but the barycenter is no longer at the centre of one of the objects. The existence of a stationary barycenter for the two-body problem is not obvious, and must be proved.
In the more general n-body problem there is no fixed point towards which the net gravitational attraction on each body always acts. So gravity can no longer be treated as a central force - this makes the n-body problem much more difficult to solve.