Declan_Basile,
>> but how does iTunes identify a computer? Does each computer have a unique id? <<
I haven't seen that iTunes publicizes this so I can't say for sure. However in almost all the cases I know of things like this are done with key hardware identifiers. Not just one piece but a few. Thus each computer's "unique" id is actually made up of a few parts of that computer. This allows a part of two to change without an issue and a better chance of a unique identifier.
The Windows key and the HD alone are not the identifiers to my knowledge.
>> If so, if my library of songs is on an external hard drive and I use the same hard drive on another computer will I have to authorize that computer? <<
Yes. The identifier isn't just on the HD.
>> If I format a computer and reinstall Windows and iTunes will I have to authorize the computer again? <<
Yes. The authorization is stored on the computer. If reformatted and reinstalled that would certainly be lost.
Let me know if you have any questions or need more information.
b0lsc0tt