Will you explain me with an example what the Damn is?
The fact that Marina's question mentioned 'the damn' suggests she's interested in the noun, so that dictionary definition isn't a lot of help - except in the example 'not worth a damn'.
The idea of 'a damn' as something of little value seems to be related to a tinker's dam (no n), a small piece of metal used to patch a leaking pot, although some dictionaries (e.g. tinker's damn: Definition and Much More from Answers.com) still accuse tinkers of swearing a lot. (Tinkers - itinerant menders of pots and pans, not usually White Anglo-Saxon Protestants - were exposed to many racist slurs.)
Anyway, whatever the derivation, the noun 'damn' can mean either something of little value ('not worth a damn') or the action of saying 'Damn' - e.g. With many a damn and blast, he finally assembled the wardrobe - or both ('don't give a damn').
The other suggested origin is the small worthless Indian coin, the Dam.