A plumbing emergency is anything actively damaging your home or leaving you without water: a burst or leaking pipe, sewage backing up into a sink or tub, a water heater failing and flooding a closet, a running toilet you cannot shut off, or no water reaching the house at all. These do not wait for business hours, so dispatch does not either: a plumber is routed to your address the same day the call comes in, any hour, any day of the year, including weekends and holidays.
If you are not sure whether what you are seeing qualifies, call and describe it rather than waiting to see if it gets worse. Whoever answers has heard most of it before and can tell you, on the spot, whether to shut off the water immediately, what to do while you wait, or whether it can hold until a scheduled visit instead.