The threshold voltage of a MOSFET is usually defined as the gate voltage where a depletion region forms in the substrate (body) of the transistor. In an NMOS the substrate of the transistor is composed of p-type silicon which has more positively charged electron holes compared to electrons. When a voltage is applied on the gate, an electric field causes the electrons in the substrate to become concentrated at the region of the substrate nearest the gate causing the concentration of electrons to be equal to that of the electron holes, creating a depletion region.
If the gate voltage is below the threshold voltage, the transistor is turned off and ideally there is no current from the drain to the source of the transistor.
If the gate voltage is larger than the threshold voltage, the transistor is turned on, due to there being more electrons that holes in the substrate near the gate creating a channel where current can flow from drain to source. This situation is called strong inversion.