A first sentence is you leaning in, close to the reader and whispering, "this is important, pay attention."
I'll second his sentiment, though I take a somewhat different view of the details than I believe paul_m_jessup does. A first sentence is a hook, but the nature of that hook can vary considerably depending on the story, the venue, the title and even (or especially) the identity of the author.
One thing I've heard a lot of in workshops over the years is a variation on "but it gets good on page seventeen!" If that's true, start the story on page seventeen. Many of us have a tendency to write our way into a story. This can manifest fairly literally as "driving to the story", a critique term for the author requiring to characters to wander about until they encounter a plot, much like a bus cruising the stops for winos. Science fiction in particular encourages this sort of behavior because there is often a need to establish the nature of the setting early on in the story. Fantasy as well, I suppose, though the reader's need to know that it is the Second Age of the Black Weasel is perhaps less critical than a clanks-when-it-walks skiffy reader's need to know what aspect of technology is in play.
Begin at the beginning is another great piece of advice, but much of the art of storytelling lies in choosing the beginning. All good stories work as "red slippers", which is to say, the characters enter the story with fully-formed lives in progress, and they leave it for other goals, assuming they survived. (The term comes from Holmesiana, where Sherlock Holmes stories often start with something on the order of "shortly after we resolved the Case of the Red Slipper," thereby establishing a continuity for Watson and Holmes.)
The first sentence sets a tone, a style, an expectation. Every rule I've ever heard for it has a boatload of counterexamples, which only leaves us with the rubric that if it works, it stands. But it needs to do something striking, no matter what.
As it happens, I also put a lot of weight on story titling. This is true whether I'm writing, editing, or reading. A good title can give a struggling story a lift. A bad title can wound a good story. The effective combination of title and first line is like a feint followed by a punch.
One the reasons the rules break down is that the nature of the first line is utterly dependent on story context. Is the piece something that builds in a slow progression? Does it explode onto the page in a burning collage of images? Is it character-driven, setting-driven, plot-driven?
To get to a specific example, one of my favorite first lines ever is in Paul Berger's "Voice of the Hurricane", in All-Star Zeppelin Adventure Stories [ Powell's | Amazon ], edited by David Moles and me.
When a herd of zeppelins comes over the horizon, you can kiss your farm goodbye.
You've got your classic character in a setting with a problem, along with a big genre flag flying there. Voice is established, the genre-specificity of the setting is established, and (at least for me), the cool factor is right up front in the phrase "herd of zeppelins." I believe this was Berger's first story, so his by-line didn't mean anything to me. With this opening sentence (and a reasonable title), it didn't have to.
Contrast with Barry Longyear's "Silent Her" [ SCI FICTION ]. The first line is:
The light was lavender, the light was white, the light was red.
That's a vague title, and a vague opening line. It sets almost nothing, bordering on the classic "white room" opening. But this is Longyear, the author who wrote "Enemy Mine" [ Powell's | Amazon ], one of the finest novellas in the history of our field. In other words, his by-line bought a full reading from me no matter how I saw the first line. "Silent Her" is an excellent story, which the reader eventually discovers is initially told from the point of view of a newborn baby — hence the unusual opening, before the child matures and her perceptions become more conventional.
Berger couldn't have gotten away with that opening line of Longyear's. Neither can I, even at this point in my career. Longyear could because of his credibility as an author. (Incidentally, that is why I firmly believe the identity of an author matters to how the story is read editorially, critically and commercially — auctorial trust is critical. Newer writers have to establish that trust anew in each story, established writers bring their credibility with them.)
Here's an exercise, if you have a little time to spend. Go find five or seven short stories you remember well now, long after reading them. For me this would include autopope's "Lobsters" and Ted Chiang's "Story of Your Life", for example. Write down the titles and first lines. Now do the same for a handful of novels.
What do these have in common with each other? Did you remember them for their titles and openers? Do the revisited titles and first lines match your memories of the piece?
Just for reference, here's the title and first lines of the last three stories I've worked on:
"America, Such as She Is" (original title, though it was written as "Untitled")
Imagine this: A small waterfront.
"Chain of Fools" (I had the title before I had the story)
Zarai examined the bucket-ship Indolent Climax.
"Human Error" (originally titled "Miners", then titled "The Bindings of Time", before I settled on the final title)
Lappet worked the mineral vein by hand.
Ultimately, write the story. The title will be what it is, the first sentence will be what it is. The more you can load there, the better, but don't force it. Stories, like wine or cheese, have to breathe.
ETA: ogre_san hits the same topic at the same time this morning, except he's smarter than me. And less wordy.