There are two different meanings of fugazi, both of them slang.
The punk band Fugazi takes its name from the term fugazi as it appears in Nam: The Vietnam War in the Words of the Men and Women Who Fought There, a 1981 compilation of reminiscences by U.S. veterans of the Vietnam War. Group frontman Ian MacKaye has explicitly said so:
I was reading a book called Nam…. A number of stories in there were kind of woven into Apocalypse Now.… It is of course filled with a lot of jargon and slang, and at some point the word fugazi… and I thought, "what is that?" And I went back to the glossary at the back of the book, and it just said "a fucked up situation." And, I thought, that was a great term!
The term is supposedly an acronym for Fucked Up, Got Ambushed, Zipped In [a body bag], recalling the older and more familiar fubar (Fucked Up Beyond All Recognition); however, it is impossible to say whether this is a backronym or its actual etymology. Other theories point to the fougasse, a type of improvised land mine. A search of Google Books turns up only the Italian surname Fugazi until the band's popularity picks up in the late 1980s.
The 2009 Routledge Dictionary of Modern American Slang and Unconventional English, edited by Tom Dalzell, also uses Nam as its example, but defines fugazi simply as
crazy. US, 1980. Coined during the Vietnam war.
A second, unrelated meaning given for fugazi or fugazy is fake or counterfeit. The word is not Italian, though a few speculate it may have originated as fugace, meaning fleeting or impermanent. According to a 2008 Word Detective post, it may have originated in famous low-budget advertisements for a tri-state area limousine rental company— the kind of outfit someone pretending to have money might hire.
The same post notes that it appears multiple times in Joe Pistone's memoir upon which the film Donnie Brasco was based, and may have been in use in the American mafia. From there, perhaps, it has drifted into pseudo-Italian slang of the sort popularized by television shows like The Real Housewives of New Jersey and Jersey Shore, and you can find fugazi in this sense now in hip-hop lyrics as well.