What would cause the cooling fan to constantly run on a Mazda 626 1999 model?

Answer: If you mean your radiator cooling fan, it most likely is a grounded or shorted circuit from the "cooling fan sensor" (internal) or a pinched wire. Depends on when it happened: was there any work done prior to this happening? IF not, then the sensor is most likely the issue. In this case, once you find it (normally the sensor is screwed into the bottom of the radiator), unplug one of the wires (or remove the connector) and see if the cooling fan continues to run, if the fan turns off, then replace the sensor.