Plain English Meaning
The lowest height you are allowed to be at when you cross the point where the final part of the approach begins.
Full Definition
The lowest altitude, published on an instrument approach chart, at which an aircraft is permitted to cross the Final Approach Fix (FAF) when beginning the final segment of the approach.
Context Anchor
Seen on instrument approach procedures and in cold-temperature correction tables when checking the altitude required at the final approach fix.
Derivation
FAF stands for Final Approach Fix — the published point where the final descent toward the runway begins. 'Crossing altitude' simply means the altitude you must be at as you pass that point.