It is not a word, it is a name, and it has two major uses. "Humpty-Dumpty" the name of a character in a nursery rhyme, quoted by Nourhan above. It is used in a situation where something is so hopeless broken that there is no chance at all of putting it back together.
A typical example of use is in a news story, in which a senator is worried about the President withdrawing the United States from the NAFTA trade agreement. The senator said that to do so would send "shock waves all throughout agriculture. And then to restitch that and put it all back together it’s like Humpty Dumpty. You push Mr. Humpty Dumpty trade off the wall and it’s very hard to put him back together."
The book title, "All the President's Men," references the poem. The situation during Watergate had blown up to the point where all the President's men couldn't put it back together again.
The second example comes from Lewis Carroll's "Through the Looking-Glass," in which a number of nursery-rhyme characters appear, expanded and elaborated on. The famous passage is this exchange between Alice and Humpty Dumpty:"
‘I don’t know what you mean by “glory,”’ Alice said.
Humpty Dumpty smiled contemptuously. ‘Of course you don’t—till I tell you. I meant “there’s a nice knock-down argument for you!”’
‘But “glory” doesn’t mean “a nice knock-down argument,”’ Alice objected.
‘When I use a word,’ Humpty Dumpty said in rather a scornful tone, ‘it means just what I choose it to mean—neither more nor less.’
‘The question is,’ said Alice, ‘whether you can make words mean so many different things.’
‘The question is,’ said Humpty Dumpty, ‘which is to be master—that’s all.’
So, "Humpty Dumpty" is sometimes used to describe situations, as in debates or in logical arguments, in which a person appears to be using words with special personal meanings that are different from what is normally understood. Someone might say "I don't understand your Humpty-Dumpty semantics here."