If one of the questions on Family Feud was ever "Name a fictional alien species", I have to think that Klingons have a pretty solid shot to be the number one answer. ("Let me see...Gungans!" BZZZZZZZT "And the Laramie family with a chance to steal!")
But why is this? Why are the Klingons so memorable?
The obvious first answer is in Star Trek's longevity. A thing doesn't remain a pop-culture concern for as long as Trek has without having some of its terminology enter the pop-cultural lexicon, and the Klingons were the main bad guys of the Original Series, so there's that. And more than that, a case could be made that the Klingons were the main alien species of the Original Series! More Klingons undoubtedly appeared on Star Trek TOS than Vulcans, and of the first ten Trek movies, there is exactly one in which no Klingons appear. (Wrath of Khan being the one – we see Klingon ships as part of the Kobayashi Maru test, but that's it.)
I think it's also easier to remember Klingons because they were often characterized as a group, more than as a race of specific individuals. Until Worf came along as a part of the Next Generation crew, we only saw a few Klingons at a time, and they never recurred (which was a shame – some of those early Klingons would have made great recurring characters, but recurring characters and stories just weren't the style back then). Klingons were, basically, the 'Indians' in Gene Roddenberry's space western that really wasn't much of a space western at all. It amazes me to this day that one of Roddenberry's concepts for pitching the show to the networks was "Wagon Train to the stars", a concept that fits Battlestar Galactica better, orders of magnitude better, than Trek. But the Klingons were the disposable villains who would pop up, make life difficult for Kirk and company, and then disappear again.
This continued in the movies, too. The main villain of The Search for Spock was Klingon Commander Kruge, who was played very nicely by Christopher Lloyd, and who might have been a decently memorable villain if not for the fact that he was following up Ricardo Montalban as Khan. There's another disposable Klingon bad guy in The Final Frontier -- I don't even remember his name, that's how memorable he was – and then, in The Undiscovered Country, the TOS-era crew finally engages the Klingons as more than villains of the week, but as complex individuals. How did that happen?
Well, in between Treks V and VI came TNG, which decided to deepen the canvas a bit when it came to the Klingons. They weren't boring villains of the week anymore; there was a weekly name attached to them. And with a spot in the regular cast, that meant installing a back story, which in turn meant finally delving into the nature of the Klingon society. TNG often did its best stuff in Klingon-culture related stories; instead of being the warlike goons of TOS, in TNG they were driven by honor more than drive for conquest. The Klingons of TNG came off as kind of a blend of Samurai and Vikings, if that makes sense. It was also interesting to me that as the Klingon Empire came to reach the limits it could reach without engaging in full-on war with other powers, their society turned its quest for honor in battle on itself, so for the rest of TNG and DS9, the Klingons were always engaged in a lot of violent interior politics and sometimes open warfare.
Of course, I can't write about the Klingons without also mentioning the physical change they went through. In TOS, Klingons were just human-looking people with demonic-looking eyebrows and uniforms of gray and black. In the opening scene of Trek: The Motion Picture, though, the Klingons were well and truly alien, with enormous spiny ridges up their bald heads. This change was never explained, really; I assume that the original thought was to just go ahead and use the film's large budget to re-do the Klingons in order to make them look a lot more alien, and that we were to just assume that the Klingons always did look like that and we were just supposed to accept the TOS Klingons as such.
But then, DS9 did its brilliant episode "Tribbles and Tribulations", which had Worf – looking like very much the spiny-head Klingon – having to be in the same room with the 'human' looking Klingons. When someone asks him about it, he simply says, "We do not like to talk about it." I'm sure this has been explained in some novel or some such, but I wouldn't know. (Heck, they may have explained it in Enterprise, but again, I wouldn't know.)
Anyway, long live the Klingons. Qapla'!