Plain English Meaning
It's how wide the localizer beam is from one edge to the other. Inside that wedge of signal, your cockpit needle tells you how far left or right of the runway centerline you are. Outside it, the needle is pegged and gives you no useful guidance.
Full Definition
The angular width of the usable signal that defines the localizer course on an Instrument Landing System (ILS). It is the total angle, measured horizontally, between the points where full-scale left and full-scale right needle deflection occur. Course width is set during installation so that the linear width at the runway threshold is approximately 700 feet, which typically results in a total angular width of between 3 and 6 degrees depending on the runway length.