Center of mass
The center of mass, abbreviated CoM, or center of gravity, is the location of an object where all mass is equally distributed around it. It is important to balance a craft to prevent it from getting out of control. Planes should have a center of mass slightly in front of the center of lift. For all craft the thrustvector should point to, point away from or go through the center of mass.
It is possible to show the center of mass in the Vehicle Assembly Building and Space Plane Hangar, but not while in flight. Also it is not possible to view the center of masses of the different stages. This is the only information about a center which is only a point and not also a vector.
Center of gravity
Sometimes, instead of center of mass, the term center of gravity is used, which describes a similar but distinct concept. Because gravity weakens with the distance to the gravity source, a long rod pointing away from the source would experience different gravitational pull at different points along its length. The "lower" end is being pulled harder than the "high" end; therefore, its center of gravity would be nearer to the gravity source than its center of mass.
In Kerbal Space Program, this difference of gravity is ignored, so the gravity is always the same on all parts of a craft, and thus both centers are in the same place.