Arguendo is a Latin legal term meaning for the sake of argument. The phrase "assuming, arguendo, that ..." is used in courtroom settings and academic legal settings to designate provisional and unendorsed assumptions that will be made at the beginning of an argument in order to explore their implications. Making an assumption arguendo allows an attorney to pursue arguments in the alternative without admitting even the slightest possibility that those assumptions could be true. Often, these assumptions would be that the facts or legal arguments endorsed by a hostile party were true.
Thus, an attorney in a criminal case may say, for example, that "assuming, arguendo, that my client stole the car, it would be clear that my client would have been justified in doing so in order to save a life." If the client would be shielded from legal consequences as a result even if he or she had committed the crime, this form of argument allows an attorney to suggest that it would be pointless to pursue the matter of whether the client committed the crime, as it would lead to the same legal consequences regardless of which set of facts was assumed to be true.
For a real-life example in a civil case, see Tiffany and Company's Reply Brief, Tiffany Inc. v. eBay, Inc., 08-3947-CV (U.S. Court of Appeals for the 2nd Circuit 2008): "In any event, assuming arguendo that requiring eBay to take remedial measures would impair eBay's business, that fact cannot relieve eBay of its legal obligations." p. 23, second paragraph.
Particularly in an appellate court, a judge may ask an attorney what the effects of a different set of assumptions, made arguendo, about the facts governing a situation might be. This is especially useful in exploring whether different fact patterns might limit the proper scope of a possible holding in a given case.