1. They have to give readers a sense of the kind of story they're reading: mood, narrative style, setting etc. Never start a book "It was a dark and stormy night" but that at least paints the required picture; you are in a dark and turbulent place or a horror satire.
2. They have to "catch" the reader to carry on past the first lines and apply the habit of reading to this work. The first sentence is best when it's a grabber. Special credit goes to Ian Banks' The Crow Road for "It was the day my grand-mother exploded."
3. They have to be both familiar and unique, so the reader can latch on to things even before the author had had a chance for the first exposition paragraphs while avoiding having the reader instantly pigeonholing the tale. "Oh, the girl's dating a sensitive vampire. Where's my shotgun?" is probably a bad reader reaction. "Oh, this is a bit like a Sherlock Holmes Victorian mystery - except the investigator is a monkey!" is probably okay.
James Earl Jones-style authorative narrative voice: "It was the best of times. It was the worst of times." That one's been used, though.
In media res dialogue: "They're breaking through! The damn mutos are already loose downstairs!" Tells you all you need to know about why the next paragraph describes our hero hefting a flame-thrower.
Paint a word picture of something's appearance or history: "Detective Brown's battered Colt '45 had a notch in the grip for every capo of the Werner gang it had ended."
Plunge in with something visceral that we're culturally conditioned not to ignore: a fight or sex are the classics, but having someone sitting there with a razorblade to the wrist, or having a woman in labour, or the flames creeping towards the baby will get the job done too. Few people aren't going to read on at least a few paragraphs to check what happens. "Lindsay Chase switched on the waste disposal grinder and slowly fed her finger between its whirling blades."
Start with a first person storyteller's plea to the reader: "Read this! You don't know what it is that lurks behind your bedroom mirror. You don't know why your reflection sneers at you when you turn away. You have to listen to me, to hear what I've seen, before the night falls and they take me - then come for you!"
Start with a word of advice. "Never trust a man selling a tonic with a hand-stencilled label." This one has a far less cerebral, far more folksy personal contact with the reader than most modern styles, but it can work if you want to set a certain tone.
Set up a paragraph that will pay off in the very last paragraph of your book to give it closure. Start with "Nobody cared that thirty people died in the 210th street tenement fire." and end with "Nobody cared that thirty people died in the 210th street tenement fire - except Rock Joe Johnson. That was why Eileen married him."
I think there's probably an art to the paragraphs right after the opening ones too, the ones that have to start serving up character and information without dumping great chunks of backstory on the poor reader; but that's probably another question.
I.A. Watson is an adventure, fantasy, and SF author from Yorkshire, England. For more information visit him online: http://www.chillwater.org.uk/writing/iawatsonhome.htm