Quantities which provide references for locations in space and time. A typical coordinate system consists of a point of reference (the origin), a set of directions (axes) that span space, and a set of labels that indicate how points are related to the origin. Coordinates in and of themselves are user defined and arbitrary, although certain simple, regular coordinate systems (e.g. Cartesian coordinates) are widely used.
A Coordinate Singularity is a location at which a particular coordinate system fails, such as the Schwarzschild metric coordinates at the Schwarzschild radius of a black hole, or lines of longitude at the North pole. This failure doesn't indicate a breakdown in the underlying geometry. It is merely a failure of the coordinate system to give a unique well-defined label to a point in that geometry.