It's an old adage in screenwriting that you have ten minutes to pull the viewer in and establish your lead characters, the tone and the "world" the story takes place in. As such, these are the ones I think do so most successfully:
Night of the Living Dead - the classic sister and brother bickering at the graveyard opening. "They're coming to get you, Barbara..." It establishes the nature of the threat the characters will be facing and Johnny's death ominously tells you that nobody in the movie is safe.
Dawn of the Dead - the chaotic TV station sequence followed by a solid 5-10 minutes of non-stop violence as the SWAT team raids the project building expertly foreshadows the even more "all hell's broken loose" finale. Everything you need to know about the movie is given to you in those first 10-15 minutes. The only thing it doesn't hint at is the humor which will come later; Romero instead opted to let the humor unfold as the message became more apparent.
Day of the Dead - the eerie "Hellooooooo?" opening punctuates the hopelessness the characters struggle with throughout the movie, not to mention it delivers one of my all-time favorite Savini zombies, Dr. Tongue.
Scream - like Az said, killing off a major star right off the bat was a shocker. Plus the sequence was suspenseful as hell and ended with a nicely gory (though trimmed for the MPAA) "money shot." The killer quizzing his victim on pop culture horror films also established the hip, referential nature of the story as well. It wanted you to know that for a change, you're watching a movie about characters how have actually SEEN horror films, and, perhaps, actually know they're in one.
Suspiria - only Argento would hammer you over the head with a violent, gory but oh-so-aesthetically pleasing double murder like this. We meet our heroine and are then shown the vicious nature of the evil awaiting her. A bravura opening sequence.
The Exorcist - the near wordless opening sequence with Father Marin in the desert establishes the heavy mood and tone of the film, establishes Father Marin's familiarity with the demon Pazuzu and shows us that Marin is old, and tired, and might not be up to the challenge he'll later face.
Alien -- most pundits refer to this flick as a haunted house in space, and it's the opening sequence which is why. Ridley Scott takes us on a journey through the ship, establishing the location, letting us see all the dark nooks and crannies perfect for some sinister force to hide in and spring out of. It also establishes a sense of claustrophobia and suggests that there is no escape.
That'll do for now. Or else I'd be on this all night. 