The opening of a story sets up the type of story we are writing. It sets up the genre: whether it's action, fantasy, crime, science fiction, romance, literary, mainstream, whatever. It does this on several levels if it's all working properly. It does this through how we start, whether it's with a description setting up the world of the novel, action, dialogue, characterisation, situation. It does this through word choice, the sound of the language, the tone of the piece, the voice.
The story opening has to do several things: engage the reader (hook them in!), establish the story problem, introduce the main character/s. I know as a writer I often think about these things. Am I setting up the story problem well enough for the reader? After all, this is something that needs to keep them interested for the whole book. Will they know what the book is about by the end of chapter 1?
I don't really think about the promise I'm making to my reader -- because it seems so obvious. Imagine you are the reader: you've picked up this nifty looking murder mystery -- the front cover is dark with a slashing blade and splash of blood. You're ready to be thrilled. The first chapter has you hooked -- there's been a murder, a grisly murder in a lonely alleyway. It's the third in a series of brutal attacks. You want to find out what happens. Then in chapter two the main character, a female PI, meets the son of the murdered man, and fancies him like mad. All her hormones are raging. The next twenty-six chapters (if you get that far) detail her attempts, unsuccessful and successful, to drag him into bed. The sex scenes are raunchy, but there's so much angst. And, er, what happened to the murder? Oh, yes, it's now become a subplot. Your gripping crime is really a romance in disguise. You throw away the book, disgusted. (I'm not having a go at romance here: you'd be equally disgusted if you were snuggled up on your couch with the latest romance novel, only to find it was really a crime novel, far more dark and violent than anything you normally like to read.)
Okay, if it's so obvious, then why am I banging on about it? Probably because it's not obvious to everyone. We all have things we don't need to think much about in our writing: for some it's their brilliant dialogue, for others it's their complex and fascinating plots, or bigger-than-life characters who leap off the page. And not all examples are as obvious as that one. Sometimes it's the formality of language, the type of diction, a particular POV. If you are going to start with first person and then jump to third after the first two pages, set these off and call them a prologue. Better still: ask yourself whether you really need to do that. Whether what you're trying to achieve offsets the likely discomfort or confusion your readers are going to experience. Usually, you're better off not being so "experimental".
If you've never thought about the promise, then it is worthwhile reading over what you're writing and thinking about it. Dig out your favourite novels and have a look at these. Could you tell what type of story they were going to be right from the beginning? My bet is you're going to answer yes.