Continuity is an expression which means there is an electrical path between two points, usually a low resistance.
Example, you are fault finding a bit of equipment, you use a multimeter set to the ohms range and you measure from the power input connector to the transformer, you get a low resistance reading.
When the boss asks what you have found you can say, "there is continuity from the mains connector to the transformer".
Many digital multimeters have a "continuity" function that emits a "beeeeeeep!" whenever you short the probes together or you connect the probes to a complete circuit. It will usually emit a beep as long as the resistance of the external circuit is under a certain number of specified ohms as outlined in the DMM's instruction manual.