Learn English – Difference between “keep something on”, “hang something on”, and “leave something on”

Collins Cobuild English dictionary:

hang up

means the same as hang: "I found his jacket, which was hanging up in the hallway".

if you hang up or hang up the phone, you end a phone call. If you hang up on someone you are speaking to on the phone, you end the phone call suddenly and unexpected.

You can use hang up to indicate that someone stops doing a particular sport or activity that they have regularly done over a long period. For example, when a footballer hangs up his boots, he stops playing football.

hang-up

If you have a hang-up about something, you have a feeling of fear, anxiety, or embarrassment about it.

hang on

If you ask someone to hang on, you ask them to wait or stop what they are doing or saying for a moment. "Can you hang on for a minute?"

If you hang on, you manage to survive, achieve success, or avoid failure in spite of great difficulties or opposition. "Manchester United hung on to take the cup."

If you hang on to or hang onto something that gives you an advantage, you succeed in keeping it for yourself, and prevent it from being taken away or given to someone else. "The British driver was unable to hang on to his lead. ... The company has been struggling to hang onto its sales force."

If you hang on to or hang onto something, you hold it very tightly, for example to stop it falling or to support yourself. "She was conscious of a second man hanging on to the rail. ... a flight stewardess who helped save the life of a pilot by hanging onto his legs. ... He hangs on thightly, his arms around my neck."

If you hang on to or hang onto something, you keep it for a longer time than you would normally expect. "You could, alternatively, hang onto it in the hope that it will be worth millions in 10 years time. ... In the present climate, owners are hanging on to old ships."

If one thing hangs on another, it depends on it in order to become successful. "Much hangs on the success of the collaboration between the Groups of Seven governments and Brazil."

hang out

If you hang out clothes that you have washed, you hang them on a clothes line to dry.

If you hang out in a particular place or area, you go and stay there for no particular reason, or spend a lot of time there. (mainly American) "I often used to hang out in supermarkets. ... We can just hang out and have a good time."

All of these, like most idioms, are for informal use.

"Hang on" and "Hold on". I often hear these words, especially on TV. People use them when they want someone to wait for something.

In the provided context, they are very similar.

Hang on:

informal wait for a short time: hang on a minute—do you think I might have left anything out? *(on the telephone) remain connected until one is able to talk to a particular person.*

Hold on can mean the same in general conversation:

[often in imperative] wait; stop:
hold on a minute, I’ll be right back!

While the end result is the same, when used as part of telephonic conversations, hold on often refers more to being placed on hold:

waiting to be connected while making a telephone call:
‘I’ll just see if he’s free,’ Rachel said, and put me on hold

As the ODO page will confirm, there are a number of other idiomatic uses of hold (hold your horses, hold your fire, hold it, etc.) which share similar connotations of stop or wait.