A comparison predicated literally on duplicity, i.e. on the shifting, imprecise, or tactical (re)definition of a linking term. For instance:
Your girlfriend lied about a relationship she had five years ago to (she rationalized her decision) protect you from hurt feelings; you lied about fucking your dog last week. You both lied and therefore are both equally to blame for your downfall (or equally blameless). [This is an example of shifting the substance of discussion via a false equivalence.]
Clinton lied about blowjobs, Bush lied about firing leakers in the Plame case, they're both liars, therefore both equally bad presidents. [This is a category mistake.]
I didn't pay you back once when you lent me a dollar, you stole a dollar from my wallet, therefore we're even. [Application of an arbitrary and wrongheaded metric - limited-scope financial equality - for 'evenness'. cf. also conversations about economic equality in general in the U.S.]
Random noise and the compositions of Mozart are equally complex. [N.B. In this case the statement is literally true by some measures of complexity, but obviously not meaningfully true for people evaluating music aesthetically. i.e. The implication, that the experiences of listening to and learning from and growing through appreciation of these two soundsets as music are somehow equivalent, is wrongheaded and silly, yet legalistically it's defensible. Put more simply/narrowly: 'complexity' and 'complication' are not the same thing, aesthetically speaking, but this statement slyly equates them - a hidden equivalence used to justify a false equivalence.]
Really, you have to extend the above definition so that negative examples are possible - i.e. making the claim of false equivalence in a false way.
So: The Soviet Union's gulags and Guantanamo Bay's 'Camp X-Ray' (and the U.S.'s use of Abu Ghraib, for instance) are incomparable because the Soviet Union was bad and we're good, therefore the linking term - 'prison camp' or 'gulag' - can't be used in an equivalence.
'If the glove don't fit, you must acquit.'
And so forth. I use the term all the time around here but it occurs to me that I've been seemingly unsystematic about it, when in fact I've had a specific definition in mind all this time. 'The teachable moment' indeed! :)