Definition
On an attitude indicator, to peg means the instrument's display has reached the mechanical or operational limit of its travel and can no longer indicate further movement in that direction. Older attitude indicators using gimbal-mounted gyros could peg (and tumble) when the aircraft exceeded certain pitch or bank limits, after which the instrument would give unreliable readings until reset.
Plain English
The instrument has hit its built-in stop and can't show any more movement in that direction, so the reading is no longer trustworthy.
Context Anchor
Seen in attitude indicator descriptions, especially when explaining the instrument face or the movement limits of its internal parts.
Derivation
From the everyday sense of a peg as a pin or stop that prevents further movement. A needle or display that has 'pegged' has run up against its stop and can go no further.
Why Pilots Care
Recognizing a pegged attitude indicator prevents misinterpretation of extreme attitudes and prompts transition to other instruments.
Intuition Check
Do not think of a peg here as a wooden peg used to hang something. In this context, it means a small fixed marker or stop on an aircraft instrument.
Example Sentence 1
During the steep recovery, the old attitude indicator pegged and tumbled, leaving the pilot to fly partial panel until it reset.
Example Sentence 2
Once the attitude indicator pegged the pilot cross-checked the turn coordinator and altimeter.