A hanging piece is a piece that can be captured for nothing: it is attacked by an enemy piece and either undefended or insufficiently defended. Leaving pieces hanging is the single most common way amateur games are lost.
When a commentator says 'the knight is hanging', they mean one thing: it can be taken, and nothing bad happens to the side that takes it. The piece is attacked, and it is not adequately defended.
The traditional French term for the same condition is en prise — older books use it constantly.
A piece can hang in two ways. The obvious one: it has no defender at all. The subtler one: it has defenders, but not enough — if a pawn attacks your knight that is defended only by a rook, the knight is still effectively hanging, because winning a knight for a pawn is a great trade for your opponent.
A loose piece is merely undefended — nobody may be attacking it yet. A hanging piece is attacked right now and can be profitably captured. Every hanging piece was a loose piece a move earlier; that is exactly why strong players track loose pieces before they become a problem. The maxim LPDO — Loose Pieces Drop Off — exists because this escalation happens constantly in real games.
Your bishop sits on g5, eyeing the enemy queen. Your opponent plays h6, attacking it. You spot a tempting knight jump on the other side of the board and play it. But the bishop on g5 now has an attacker and no defender — it is hanging, and it disappears next move.
Nothing about that sequence involved deep tactics. The bishop was lost because attention moved elsewhere, which is how most pieces are lost.
Almost nobody hangs pieces in puzzles, because puzzles announce that something is going on. In real games there is no announcement. The only reliable protection is a pre-move safety check run on every single move: What did my opponent's last move attack? Which of my pieces are undefended? What does my planned move stop protecting?
Players who run that check consistently stop hanging pieces within weeks. Chessdock helps the habit stick by finding the positions from your own games where you hung something and serving them back as puzzles — the same board, the same trap, until you see it coming.