Center of thrust
The center of thrust (abbreviated COT or CoT) is the midpoint where thrust from all a craft's reaction engines balances and the direction in which a craft's thrust is acting. It is often called the thrust vector, though this may emphasize the direction at the expense of the position.
Along with the center of mass (COM), and the center of lift (COL) when inside an atmosphere, it is one of three key physics properties determining a craft's behavior while in motion. Display of the COT can be toggled in the editor where it is seen as a magenta and black checkered sphere with a magenta arrow extending out of it.
Like lift, thrust is a force acting on a craft and the center of mass acts as the fulcrum. The center of thrust has a vector arrow pointing the direction that thrust will push out of the COT, causing the craft to accelerate per Newton's Third Law. The interaction between the COT and COM is charted along the axis running through the exact center of each of them. For the craft to travel straight and true along that line, the thrust vector points the exact opposite direction along the axis from where the craft is going (equal and opposite reaction).
If the vector of thrust tilts at an angle to that line, the result will be asymmetric thrust which generates turning torque around the COM. This is how thrust vectoring works, but torque steering can be excessive due to design. For example, if the engines on a plane are above the COM, they may point straight back but the COM-COT line doesn't; thus the thrust vector is tilted relative to it and the plane will nosedive. Such torque can sometimes be balanced out by control surfaces, RCS, or SAS, but it is more fuel efficient to make a balanced craft with its COM in-line with its COT. And the more thrust the engines generate, the stronger any torque steering becomes.
It is important to note that the center of thrust depends on all running reaction engines on the craft. Unfortunately, the editor calculates the COT based on the thrust of all engines present, regardless of staging. In use, the COT will change when any single engine's thrust changes due to activation, throttling, shutdown, or any other change. This includes when a craft's configuration changes due to jettisoning an engine or docking with another craft that has an active engine.
A highly unbalanced rocket (e.g. a rocket that loses one of multiple engines due to rapid unplanned disassembly) might produce so much torque at full throttle that its other controls are too weak to prevent it from rotating. This means it will be mostly uncontrollable. It might be possible, however, to control such a craft by throttling the engines down to a point where the torque they produce is no greater than the torque from SAS.
Pendulum rocket fallacy
It is tempting to think about placing the engines of a rocket at the top, so that the thrust vector points down toward the center of mass. This way, it seems that the engines will pull the rocket up, and that the rocket's weight will keep it hanging directly under the engines, so that it stays upright. Unfortunately, this is not true. Since the rocket's weight acts as a force exactly through its center of mass, gravity doesn't provide any torque to oppose the rotation of the rocket. Any imperfection in the alignment of the thrust vector, or any outside torques, will still cause rotation. This is equally problematic regardless of the location of the engines. The stability of a rocket ultimately depends on its ability to actively counteract imperfections. In KSP, this is most often accomplished by using engines with thrust vectoring. When such an engine gimbals, it slightly modifies the thrust vector. SAS uses this as needed to correct for imbalances, along with RCS, reaction wheels, and control surfaces.