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A musician friend mentioned this in passing - she talked about the something chord and then explained it was unresolvable, and how she hates it because she cannot stand music that doesn't resolve.
But I can't recall what it was called. I don't know enough to know if it really is a single chord, or a family of chords, or something less specific entirely.
It sounds like she may have been talking about the Tristan Chord, a famous chord from the opening of Richard Wagner's opera Tristan und Isolde.
While it can be enharmonically written as a half-diminished 7th chord (F-A♭-C♭-E♭), it does not resolve in the way a half-diminished 7th chord would, nor is it written as a half-diminished 7th chord. For this reason, it has become famous for being a rather difficult chord to analyze with traditional Roman numeral analysis. The issue is not that it doesn't resolve, but that it resolves in a way that makes its relationship to the key so ambiguous.
I'm not sure I could call any chord "unresolvable", though I'd have to know the context of the conversation.
However, augmented chords (e.g. C-E-G♯), as well as diminished 7th chords (e.g. C-E♭-G♭-B♭♭) -- both of which have been mentioned in other answers -- share a common trait: they are, in some sense, symmetric. Augmented chords are created by stacking three major thirds, and diminshed 7ths are likewise created by stacking four minor thirds, both of which add to an octave. While neither chord is "unresolvable", these two types of symmetric chords could be said to be "ambiguously resolvable", in that they look the same from three or four different keys (respectively), and can therefore be plausibly resolved to any one of several possible distant keys. As a result, romantic composers will often use them as a sort of "turn-stile" to modulate between unrelated keys.
In fact, the Tristan Chord, mentioned in yet another existing answer, is also somewhat symmetric (at least after the G♯ resolves to the A). It consists of two major thirds (F-A and B-D♯), separated by a tritone (a symmetric interval). As such, you could almost envision it resolving to a Bb7 instead of an E7 (if it weren't for those chromatic passing tones that help to indicate a clear direction to the line).
The takeaway point is that, in music, symmetry leads to a certain ambiguity of tonal center, and therefore can weaken the sense of needing any specific resolution. Such a chord still needs a resolution, but it has several options, and so does not necessarily need any specific resolution.
They may have said the Tristan Chord, but I would argue it is resolvable, and in fact not mysterious... as others have pointed out, the analysis works well if you see the G# as a non-harmonic tone, a lower neighbor (in A minor) to the A at the end of the measure.
Then we have the following notes, spelling them out in thirds: (B D# F A). We have F in the bass. This is a textbook French 6th chord in A minor!
http://en.wikipedia.org/wiki/Augmented_sixth_chord
Note that the next chord is V7, a dominant chord in A minor, which is exactly what you would expect (perhaps with an intervening I 6/4 but not seen here) following the Fr6.
As I recall Wagner loved augmented sixth chords. I've not done any research on the Tristan Chord specifically before seeing it here, but I'm surprised it has an aura of mystery.
Personally, I find augmented chords (like c e g#) much worse to resolve than diminished chords: diminished chords reduce to a seventh chord by lowering any chord note by a semitone and resolve obviously from there. One can often actually use them functionally instead of a seventh chord in the first place.
Augmented chords don't work in that manner. If you change one of their notes, the result will be a finished chord, but one that is not in useful harmonic relation to the augmented chord and thus does not serve as its resolution or as a satisfactorily meaningful part of a multi-step resolution.
Consequently, if you take a look at resolution master J.S.Bach, you'll find diminished chords all over the place in cadenzas, but mostly draw a blank on augmented chords.
While Dom talks about the V chords which in themselves sound unfinished, they are resolvable. I think it may be diminished chords that you have in mind.They use notes which are usually not within the key in question, thus sound a little strange, and can, by their make-up, go to several different places. The V chord, 9 times out of 10, will find resolution in the I chord. Everywhere in Western music.
