Hung and hanged are both past forms of hang, but they are not used the same way.
In modern English, hung is the usual choice for things being suspended or placed in a hanging position. This applies to objects and to people in nonexecution contexts.
- The coat hung by the door.
- A lamp hung from the ceiling.
- He was hung upside down in the stunt.
Hanged is generally reserved for execution, especially in legal, historical, or formal writing.
- The prisoner was hanged in 1850.
- He was sentenced to be hanged.
A simple way to remember it is this: if you are talking about an object, decoration, or ordinary physical position, use hung. If you are talking about execution by hanging, use hanged.
This distinction is one of those small grammar points that can make your writing sound much more precise.