"Resistance is futile" is one of two catchphrases used by the Borg in the Star Trek franchise, spoken with the sort of ominous finality the Borg are well-known for. They leave no room for options, and they know that they have overwhelming force on their side. They are powerful, adaptable, and relentless, and they will win in the end, no matter how long it ultimately takes or how many temporary setbacks they encounter. They will never give up, they will adapt to defeat any weapon or tactic used against them, and they will eventually overwhelm and conquer.
Despite its immediate association with the Borg, they are not the first to use the phrase, and they didn't even use it themselves until their second on-screen appearance. In their first appearance, Q Who? (1989), the closest they come is "If you defend yourselves, you will be punished." It's not until The Best of Both Worlds, Part I (1990) that they tell Picard "Resistance is futile" before they implant the cybernetic parts that turn him into one of them. Picard, now Locutus, then passes the message on to the USS Enterprise-D.
Wikipedia reports that the exact phrase was first used in Space: 1999 in 1977, and later in an episode of Doctor Who in 1983. Long before then, however, the remarkably similar phrases "resistance is useless" † and "to resist is futile" were used by the Doctor Who enemies the Cybermen (themselves in some ways similar to the Borg) as far back as 1967. Indeed even before that, the Daleks told The Doctor that "resistance is useless" in 1964's The Dalek Invasion of Earth‡. Despite that, however, the Borg were clearly the most popular sci-fi villains to have used it and brought the phrase into the mainstream, going so far as to use it as the tagline for Star Trek: First Contact.
† XWiz and The Custodian point out that "RESISTANCE IS USELESS!" was also the catchphrase of the Vogons of The Hitchhiker's Guide to the Galaxy. This was likely a direct reference to the Doctor Who villains, as Douglas Adams was involved with Doctor Who around the time THGttG made its first appearance as a radio play in 1978.
‡ Thanks, Master Villain.
In everyday usage, both "Resistance is futile" and the Borg's other, equally ominous catchphrase "You will be assimilated", are used in reference to organizations with monopolistic tendencies, particularly Microsoft. Famously, in the early days of amateur digital picture manipulation's popularity, Borg implants were added to a headshot of Bill Gates, and is still used today as an icon on Slashdot to mark news articles and discussion about the software giant.
Alas, the very unstoppable nature of the Borg was their own undoing, at least so far as the Star Trek franchise was concerned. Because resistance was, in fact, futile, the writing staff must have had one heck of a time coming up with ways to defeat them, and they only appeared in a half-dozen episodes of Star Trek: The Next Generation (four if you consider that two of them were two-part episodes).
For their first big-screen appearance in Star Trek: First Contact, the Borg were changed dramatically, with the addition of nano-robot injections that quickly converted victims into more Borg and the ridiculous Borg Queen, which I feel defeated the entire purpose of the robust, massively redundant Borg organization and gave them a heart to stab at. Further appearances in Star Trek: Voyager almost reduced them to a joke, as a single, lost Federation starship got the better of them time and time again. Resistance didn't turn out to be quite so futile after all.
Sources:
http://memory-alpha.org/en/wiki/Q_Who%3F
http://memory-alpha.org/en/wiki/The_Best_of_Both_Worlds
http://en.wikipedia.org/wiki/Resistance_is_futile