If you have access to software or electronics that slows the music down, then you can record the chord in question, and isolate it so that it plays over and over. Then you can pick it out note for note until you get a precise match for your ear.
If you can't slow it down to do that, then you have to keep replaying the music over and over on the cassette/CD/turntable, and do essentially the same thing. I start with the bass note, or root of the chord first, trying to get a match for the root. That at least tells you what the root chord might be, for example a C-something. Once that's decided then you can try all variations of the C chord that you know (e.g., C7, Cm, Cm7, Cmaj7, C6, etc etc), playing them along with the music, hoping that you come up with a match. Only thing is, you have to be pretty familiar with a lot of chord variations.
Not very scientific I'll admit....but it works for me - most of the time - as long as the chord isn't some weird dissonant jazz chord.