There are at least two ways to interpret "the relativist fallacy": either as identical to relativism (generally), or as the ad hoc adoption of a relativist stance purely to defend a controversial position.
On the one hand, those discussions of the relativist fallacy which make the fallacy out to be identical to relativism (e.g., linguistic relativism or cultural relativism) are themselves committing a commonly-identified fallacy of informal logic, namely, begging the question against an earnest, intelligent, logically-competent relativist. It is itself a fallacy to describe a controversial view as a "fallacy"--not, at least, without arguing that it is a fallacy. In any event, it will not do to argue as follows:
On the other hand, if someone adopts a simple relativist stance as an ad hoc defence of a controversial or otherwise undefendable position--saying, in effect, that "what is true for you is not necessarily true for me," and thereby attempting to avoid having to mount any further defense of the position--one might be said to have committed a fallacy. The accusation of having committed a fallacy might rest on either of two grounds: (1) the relativism on which the bogus defense rests is so simple and meritless that it straightforwardly contradicts the Law of Non-Contradiction; or (2) the defense (and thus the fallacy itself) is an example of ad hoc reasoning (which see). If a position cannot stand up to logic, then one can still maintain that the position is correct by asserting that truth, or standards of logical consistency, are relative to a particular thinker or group, and that under some other logical standard, the position is correct.
On any interpretation of the fallacy, in determining whether the relativist fallacy has been committed, one should distinguish between things which are true for a particular person, and things which are true about that person. Take, for example, the statement profferred by Jim, "More Americans than ever are overweight." One may introduce arguments for and against this proposition, based upon such things as standards of statistical analysis, the definition of "overweight," etc. It is a position which answers to objective logical debate. If Joe answers Jim, saying "That may be true for you, but it is not true for me," he has given an answer which is fallacious as well as being somewhat meaningless in the context of Jim's original statement.
Conversely, take the new statement by Jim, who is 5'6" tall, "270 lbs is grossly overweight." Joe, who is a foot taller at 6'6", and weighs an exact, well-conditioned 270 lbs, replies, "That may be true for you, but it is not true for me." In this context, Joe's reply is both meaningful and arguably accurate. As he is discussing something which is true about himself, he is not barred from making an argument which considers subjective facts, and so he does not commit the fallacy.
Some opponents of Marxianism identify, as having committed and propagated the relativist fallacy, Karl Marx, who argued that different social classes have different logics, and that the arguments refuting his theories were merely bourgeois logic, and as such could be dismissed without needing to be refuted. Marxianss would reply that it is a fallacy to define, without argument, their relativism--called polylogism--as a fallacy.
If one tries to translate "That may be true for you, but it is not true for me," into E-Prime (meaning English without the verb 'to be') one can choose from a number of possible translations. Some of these seem perfectly self-consistent. For example: "That may follow logically from your arbitrary set of definitions, and you may think you remember empirical evidence for your position or even that you perceive such evidence right now, but I strongly doubt you can convince me of what you say."