1.

There are actually two quite different meanings
for hang and these meanings have different forms:

a. hang / hung / hung

b. hang / hanged / hanged

2.

One of the common meanings for hang is
"to suspend." The forms for this meaning are
hang / hung / hung.

Examples:

Where can I hang my coat?

Who hung that picture in the living room?
It's crooked!

They've hung stockings on their fireplace
every year at Christmas. It's a family tradition.

 3.

The other common meaning for hang is
"execute by hanging" (using a noose). This
meaning is generally used in passive voice.
Its forms are hang / hanged / hanged.

Example:

The murderer was hanged as punishment
for his crimes.

Hang is also used with head to mean
"incline to a lower position" or "bow":

He hung his head in shame.

There are several common phrasal verbs
and idioms using hang:

be hung up on / about (be fixated or
unnaturally worried about something)

hang around (wait)

Hang in there! / Hang on! (Don't give up!)

hang on (wait; survive despite difficulties)

hang up (replace the receiver during
a telephone call)

hang-up (noun: unnatural or extreme
worry about something)

All of these expressions (except the noun
hang-up) use the forms hang / hung / hung.
The forms hang / hanged / hanged are used
only for meaning #2 ("execute by hanging").