Definition of "hanging a piece"?

People often talk about how they hung a piece. What exactly does this mean?

If I miss that my bishop is en prise and undefended, I obviously hung it...

What about if my opponent forks my king and bishop because I missed the threat? Have I still "hung" the bishop?

What if my opponent attacks my rook, forcing me to move it, which allows him to fork my king and bishop next move? Have I still "hung" the bishop?

And so on.

Yes and no. 
If your piece is undefended then its surely hanging a piece.
If some form of combination is involved in the capture of the piece then its not technically hanging a piece. 

I do consider a fork hanging your piece though...

Sorry just new. What does "forking" mean ?

I think it's when you position a piece in such a way that you attack 2 of your opponents pieces and they are forced to give up at least one of them? Like if a knight checks the king but can also attack the queen, you fork then because king is forced to move and you can take the queen. Someone might explain it better than me xD

The White knight forks the Black queen and the Black rook. The Black queen forks the White bishop and the White rook.

In this case, the White knight will capture one of the attacked Black pieces, but the same White knight can be recaptured by the other uncaptured piece. So technically there is no hanging piece available for White though.

The following is also technically a fork.

Get the piece on a scaffold, put a noose around his neck, then pull the lever.

 A piece is hanging is more it looks if it is save or defended but it is not. becuase a lot of pieces are on sand that is lose. After some capures a piece can be hanging because it was before defended but because of the changed situation not anymore. This can be for example because of a pin of a pin that does not works anymore. Or a pawn is undermined that was an importent defender of a knight. Then suddenly a piece can be still hanging and will be lost. I think it means something like this.

Isn't it "hanging a piece" and "forked" are two different situation? 

"Hanging a piece" is a situation where an undefended piece can be taken without an exchange

whereas, "Forked" is a situation where a higher value of material will be taken with or without an exchange.

I think, normally a "Knight" would do the trick on a fork. My question is.....

Can  any piece (beside King of course) including pawn be considered a fork with above definition? i.e a Pawn attacking two major pieces, Bishop attacking K and Q? Or does it fall more on "Hanging"?

A fork is a type of double-attack, where one piece/pawn threatens two or more opposing pieces.  All forks are double-attacks, but not all double-attacks are forks.

A hanging piece is one that can be taken for free.  Any piece that is not defended by another piece is considered loose. There are also cases where a piece is defended but not sufficiently defended, like below.

If a player has several loose pieces, we often say the whole position is loose.  Spotting such loose pieces is often the first step to spotting tactics.

Heard some muttering in the playing hall.  "I'm all forked up.  Hung my piece again."

A hanging piece is usually simply defined as an undefended piece.

But I supposed a clearly underprotected piece can also be considered hanging.

And what about X-Ray attacks?  Is the bishop on b7 below hanging because it's only protected once but attacked twice: attacked directly by the rook at b5 and indirectly by the queen at d5?  When Black moved his king to h8, it sure seems like "he left his bishop hanging."

Personally, I've examined my success at tactics problems over at chesstempo, where my tactics rating is 1760, and see that I've only solved 37 of 64 problems, with a 1573 performance rating in problems with X-Ray attacks as a key part of the solution.

When I look for hanging pieces and where I can set up double attacks, I see that I'm neglecting to look for attacks that GO THROUGH other pieces.  So I'm trying to keep X-Ray attacks always in mind: