hang and its derivative words have various uses in chess parlance.
To "hang" a piece is to allow it to be captured "for free" (i.e., without the possibility of an equalising recapture on the next move). The word "hang" is usually employed when such a situation occurs because of a silly error; deliberately allowing a piece to be captured would instead be termed a "sacrifice". Example usage: "Fischer overlooked a simple tactic in the early middlegame and hung his bishop".
An insufficiently protected man, one liable to be hung, is said to be "hanging". This may be because the piece is completely unprotected (in this case it would be described as "hanging" even if no enemy men are currently attacking it), or protected by fewer pieces than the number attacking it, or if it is protected only by a pinned piece. Example usage: "The knight on c4 is hanging, so White needs to prevent the black queen from coming to b4 with check, winning a piece".
Another usage appears in the term "hanging pawns", which describes a pair of like-coloured pawns on adjacent files flanked on both sides by files containing no friendly pawns. For example, if White has pawns on c4 and d4 but no pawns on the b- or e-files, then the c- and d-pawns would be described as "hanging".