Plain English Meaning
The flight director (or autopilot) locks onto the runway alignment signal and steers the aircraft to stay on the extended runway centerline as it flies the approach.
Full Definition
A flight director or autopilot mode in which the aircraft automatically follows the localizer signal of an Instrument Landing System (ILS), keeping the aircraft aligned with the runway centerline during an instrument approach.
Context Anchor
Seen when using a flight director system during an instrument approach that includes a localizer.
Derivation
Localizer comes from 'localize' — to a precise location or line. The localizer defines the line you must stay on; means following that line continuously. Together: following the line that locates the runway centerline.