It's a phrase used by the Borg as part of their standard greeting, the idea being to scare potential targets into acquiescing with as little expenditure of Borg resources as possible.
In 'Best of Both Worlds", Picard tells the Borg that he has nothing to say to them and that he will resist them with his last ounce of strength.The Borg reply: "Strength is irrelevant. Resistance is futile. We wish to improve ourselves. We will add your biological and technological distinctiveness to our own. Your culture will adapt to service ours." Picard tells them that his culture is based on freedom and self-determination. The Borg inform him that: "Freedom is irrelevant, self-determination is irrelevant. You must comply." Picard retorts that he would rather die. The Borg reply: "Death is irrelevant..."
Thanks to the popularity of that episode, the phrase "Resistance is futile" became the slogan for the Borg. By the way, the Borg in this episode and Patrick Stewart both say "fu-tile". Whereas Data in First Contact pronounces "futile" like most people would say "feudal". Admiral William Thomas Riker 22:06, July 26, 2011 (UTC)
Except that it hasn't been futile at all, at least not for the Federation in virtually every encounter (canon or apocrypha). It has been costly, however...
More than that, "Resistance Is Futile" has made its way into the larger American cultural lexicon, to refer to the assimilation of any smaller group by a larger group. For instance, the corporate commercial activities of Microsoft either buying or building its own computer and internet properties in relation to the startups that originated them. ProfessorTrek 00:19, July 27, 2011 (UTC)