A mathematical vector used to describe the distance and direction between the origins of two separate coordinate systems. Its coordinates are used to 'translate' the coordinates of a point (positional vector coordinates) in one coordinate system (CS) into that point's coordinates in a different CS that has a different origin.
If the coordinate systems are identical with the exception of the origin locations, the translation vector's coordinates are added directly to the points initial coordinates to yield the point's coordinates in the final coordinate system.
See also Rotation Matrix and Homogeneous Transformation Matrix
For detail and illustrations see the tutorial:
REFERENCE COORDINATE
SYSTEMS AND TRANSFORMATIONS