What '[on] fleek' means and where it came from: the standard view
As Matt Эллен notes in his answer, the road to mass adoption of fleek runs through a Vine mini-video uploaded on June 21, 2014, by Peaches Monroee. If you don't have Flash on your computer (as I don't), you can relive Ms. Monroee's 12 seconds of stardom here on YouTube. One commenter at YouTube conveniently transcribes the audio of the mini-video as follows:
We in dis bitch, finna get crunk; eyebrows on fleek, dafuq?
The suburban, red-state translation is roughly this:
We are in the car, fixing to have a wild time; my eyebrows look absolutely fabulous, so what the heck?
A number of websites have dedicated articles to answering the two-headed question, What does fleek mean, and where does it come from? The nature of the Internet being what is (namely, a big echo chamber), most of the most heavily trafficked articles reach very similar conclusions. From Know Your Meme ("Eyebrows on Fleek"):
The earliest known definition of the term “fleek” was submitted by Urban Dictionary user Dan Blue on October 2nd, 2003, defining the term as “smooth, nice, sweet.” On December 1st, 2009, Urban Dictionary user Alycyn submitted another entry for “fleek,” defining it as a synonym for “awesome.”
Know Your Meme doesn't hazard an opinion of its own about what "on fleek" means, but by way of context it notes that Peaches Monroee uses her Vine mini-video "to show off her stylishly groomed eyebrows." Clearly, "on fleek" is a good thing.
From Four-Pins.com "Twitter On Fleek: Corporate Memes, Joke Attribution And The Life Cycle of Cool," October 8, 2014):
"On fleek" is a cooler way of saying "on point" and draws its origins from a Vine made by Internet hero Peaches Monroe in late June of this year. ... As far as I can tell, "on fleek" was popularized by Monroe herself, though Urban Dictionary user Dan Blue was out here on the saying that "fleek" meant "smooth, nice, sweet" all the way back in 2003.
From Racked.com ("A Brief History of Fleek, From Taco Bell to Kim Kardashian," November 5, 2014):
Put very simply, fleek is just another way to say "on point." And if your eyebrow game is on lock, your eyebrows are on fleek.
Over the summer, Monroee's Vine introduced the world to "fleek."
From Uptown Magazine ("Trademarks on Fleek: Who Owns the Word?," November 18, 2014):
[L]ast week while attending a B.O.B. concert I was equally confused when an artist from his No Genre label came on stage singing about her eyebrows being on fleek. I turned and asked aloud, “What is a fleek?” Apparently, “fleek” means “on point” and “on the mark.”
‘Fleek’ is quickly gaining traction as the new ‘it’ term in sports, fashion, music and social media. Unbeknownst to many, the term was first written in the 1st Century. According to a 1801 translation of Plutarch’s Lives, the term first appeared as follows: “fleek fellows I am afraid of, but the pale and the lean.”
Unfortunately for the Uptown Magazine author, the instance of "fleek" in the cited 1801 edition of Langhorne & Langhorne's translation of Plutarch's Lives is in fact an instance of "ſleek"—that is, "sleek"—which the author could have confirmed with a quick trip to Google Books search (the occurrence of "theſe" two words earlier and of "Caſſius" 18 words later would have provided useful clues as to what was going on with "ſleek"). I'm also at a loss to explain why the author considers 1801 to belong to "the 1st Century."
From Billboard ("From 'Pulling a Beyonce' to 'On Fleek,' Slang Terms That Invaded 2014," December 12, 2014):
Meaning amazing, impeccable; originated by Vine user Peaches Monroe (to praise her eyebrows) and popularized by Ariana Grande, who sang it on MTV in August.
From Betches Loves This ("On Fleek: Betchy Linguistics," December 22, 2014):
Before you ask me what “fleek” by itself actually means, I have no fucking clue.
In terms of what it means, it’s actually really easy and hopefully you figured it out for yourself already. But for all you Karen Smith’s out there, on fleek = on point. Like literally, they’re interchangeable terms, but one of them makes you sound slightly more edgy.
From Bustle ("What Does "On Fleek" Mean? A Brief Timeline of the Phrase that No One Really Understands, But Everyone Keeps Saying Anyway," December 29, 2014):
I’m not sure when the first time was that I heard someone describe something as ”on fleek.” ... What is truly confusing about the phrase itself is the fact that fleek is not a word. And it doesn’t even really sound like a word. It just… exists. And regardless of whether it’s a word or not, it’s cool. Granted, maybe most people are using “on fleek” ironically rather than seriously, but still… it’s trendy. And I, for one, would just like to admit that I have no idea what the heck it means. Where did it even come from?
1. The Origin
The earliest mention I could find of the term was described by People as being listed on Urban Dictionary as early as 2003. According to that entry, “on fleek” meant ”smooth, nice, sweet.” However, the term really didn’t pick up in popularity until this year, 2014.
2. The Meaning
As you can see from the video [of Peaches Monroee], and general use of the word, it means that something is on point, essentially — that something looks good, is perfectly executed, etc. It’s just another way to compliment someone else… or yourself. Why not?
Source theory #1: 'Fleek' ultimately derives from an Urban Dictionary definition posted in 2003
Here is the famous entry for fleek posted by Dan Blue on Urban Dictionary on October 2, 2003:
fleek smooth, nice, sweet [Example:] "That was a fleek move you pulled on that chic."
The first thing you may notice about this definition is that it uses fleek as an adjective—not as a noun, and not in tandem with on. The next thing you may notice is that it is not alone. Aside from a small landslide of post–June 21, 2014 offerings, UD enthusiasts have suggested an array of definitions for fleek over the years. One of these, posted by Alycyn on December 1, 2009, might be seen as a missing link between Dan Blue's 2003 definition and Peaches Monroee's 2014 usage:
fleek English slang means awesome [Example:] That was a fleek game.
But you'll also see a gaggle of less promising entries on UD. For example, posted by AKJenk on July 4, 2005:
fleek Sure, men may never experience the pain of child birth, but women will never fully appreciate the agony of a fleek. A fleek is a light flick to the testicles. [Remainder of definition omitted.]
And posted by xxreptarlampxx on December 7, 2010, we have this:
fleek v. to sing the song of one's people in a traditional manner, usually while dressed in ancient garments.
And posted by birthmas on December 11, 2010, we have this:
Fleek A marionette that is frightening and haunts you in your dreams.
Not to mention inane definitions for Fleeke (from January 23, 2008), fleeked (from October 19, 2006), and fleeko (from January 11, 2011), along with another "awesomeness"-related word, fleeky, posted on August 3, 2007:
fleeky a word meaning awesome, splendid, cool, happy, great, outstanding, etc. [Examples:] 1. He is soo fleeky! 2. That concert was definitely fleeky!
A competing definition of fleeky posted on January 21, 2012, however, asserts that "fleeky is an onomonopia name, it is what it sounds like it. the name was created by a musician named 0110 who was apart of an experiment music group called Communion in 1996 somewhere in the east coast of the U.S.A. "
It is interesting that three of these definitions involve positives ranging from "smooth, nice, sweet" to "awesome"—but before I took that as strong evidence that Dan Blue and Alycyn were talking about the same fleek that Peaches Monroee was, I'd want to know how many of the thousands of imaginary words posted on UD also allegedly mean "awesome." I note, for example, that a poster on July 12, 2011, claims that Flege means "A person who is extremely awesome, amazing and beastly"; a poster on February 16, 2008, reports that fleischut means "amazing or the best thing ever in the whole world"; and a poster on April 13, 2008, reports that flemmin means "cool, rad, or awesome."
Ultimately, Urban Dictionary is so full of random inventions that it is impossible to tell whether a definition offered eleven years ago that happens to be reasonably close to the current definition of a word spelled the same way has any ancestral connection to the current word. At least with regard to fleek 2003 and fleek today, I'm skeptical. And if the two are directly related, that information doesn't bring us any closer to an understanding of why fleek was used in the first place.
Source theory #2: 'Fleek' is a variant of 'flick'
From PopSugar ("Eyebrows on Fleek Were the Biggest Beauty Trend of 2014," December 27, 2014):
Fleek — which is a variation of "flick," a word well known to makeup-lovers — is when eyebrows are perfectly groomed, filled, and shaped. In essence, "brows on point" and "brows on fleek" are the same thing.
Well, this is different. The PopSugar author says that makeup-lovers are very familiar with the word flick—so I suppose that the way to test her assertion is to look for instances where flick comes up in the context of cosmetics and try to nail down a definition of the term.
One of the first things I discovered in pursuing this question was that flick is indeed a common term used in connection with eyeliner. One online tutorial promises to show you how to perfect your "feline flick" for cat eye makeup; another article reveals "7 Hints for Creating the Perfect Eyeliner Flick." In fact, dozens of sites discuss the "flick," which refers to the "winged eyeliner" marks on the outer edge of each eye—a less exaggerated version of the eye makeup style that Amy Winehouse used to sport.
But these flicks involve the eyelashes and eyelid, not the eyebrow. A Google search did lead me to a few interesting references to "eyebrow flicks." One is from "Eyebrow Lift" at Skin Beautiful Medical & Cosmetic Clinic:
An Eyebrow lift (brow lift, brow flick, Nike tick, brow levelling) is an advanced non surgical cosmetic procedure that raises the position and/or perspective of the eyebrow relative to the eye and forehead.
It seems highly unlikely, though, that Peaches Monroee was discussing an advanced nonsurgical procedure that she had done on her eyebrows at a cosmetology clinic.
Another mention occurs in Laura Bradley, "Women with Thick Brows - In Pictures," in AnOther Magazine (August 19, 2011):
Since the moment AnOther Magazine's spring/summer 2011 cover came in, featuring Mia Wasikowska shot David Sims, I've been dedicated to thick brows. The infatuation actually first started when I saw Mariel Hemingway as Tracy in Woody Allen's 1979 film Manhattan. Playing a 17-year-old impressionable girl that Allen's character Isaac Davis is dating, she appears in the film wearing very little make-up, if any. Her natural dark eyebrows, unique for their flick at the end, are one of her striking and most enduring features.
The included photo of Ms. Hemingway from a diner scene in Manhattan shows that her left eyebrow is indeed avec (if not on) flick. Ms. Monroee's eyebrows, however, lack any such flick.
And a third mention is in "Fascinating Eyebrow" on All the Tropes Wiki, which uses "eyebrow flick" synonymously with "temporarily arched eyebrow":
Seven of Nine from Star Trek Voyager would arc her ocular implant when the occasion required, of course, said implant was always in this position so it'd be hard not to... She even fought Dwayne Johnson in a Forced Prize Fight episode, with The Rock doing his famous eyebrow flick to the audience.
So is "eyebrows on fleek" simply a variant of "eyebrows on flick"? Maybe. But even if it is, I can't tell whether being "on flick" originally meant "on point," "well exercised," "well groomed," or something else. To its credit, this second theory ties fleek to a preexisting word instead of conjuring it out of thin air. But the evidence for fleek = flick is far from dispositive.