Posture most commonly refers to keeping your head up and spine upright while inside the guard. If the bottom player breaks your posture — pulling your head down to the mat — you are close to a triangle, armbar, or guillotine. Maintaining posture means pushing up with your arms, keeping your hips forward, and not letting them control your collar or head.
From the bottom, breaking your opponent's posture is the first step in most attacks from closed guard. You use a cross-collar grip or behind-the-head control to bring their weight forward, which shortens the distance, controls their balance, and opens submission entries.
Posture matters in other positions too. In side control, staying heavy and posting a knee blocks the bottom player's frame. In standing, a collapsed spine makes you easier to throw.