[This answer is a community wiki. Please contribute any other interesting and relevant articles or examples to list at the bottom.]

In a slightly different vein to answers so far: don't approach people first, just shoot them. This is mostly for practical reasons; you don't get good street photography by asking permission first (though you will get some great portraits that way).

Some examples in action:

• Gary Winogrand (briefly): http://2point8.whileseated.org/2007/03/23/garry-winogrand-with-bill-moyers/
• Joel Meyerowitz, discussing shooting street specifically: http://www.joelmeyerowitz.com/photography/interview_06.html
• Jeff Mermelstein (briefly): http://2point8.whileseated.org/2007/08/01/jeff-mermelstein-video-from-media-matters/ (excellent contrast to Meyerowitz!)
• Daido Moriyama: http://www.youtube.com/watch?gl=FR&hl=fr&v=VViYphLJWOk

Something you will see in common in these: they are all at ease. Even Meyerowitz bobbing up and down excited about the dogs - he's relaxed. They keep their cameras in their hands, raise it, frame and take the photo quickly, and lower the camera. Then they engage, briefly, and usually non-verbally. Even the quiet and contained Jeff Mermelstein gives a smile or a nod. This isn't impolite or aggressive, it's just a different set of mannerisms and timing.

In as few words as possible: Take the photo quickly and without fuss, then look at the subject, smile happily, and mean it.

That's where you engage your subject, and reassures them this is all totally normal and that you're not doing anything sneaky. It happens quickly, so it's not a big deal. This type of body language, communicating comfort and calmness, works incredibly well. It does not take as much practice as you might think, and you need not be the social butterfly extravert type; again, just look at Mermelstein!

In my several years of shooting street and trying to emulate this approach, the most usual reactions I've seen, in order:

1. Nothing/vaguely puzzled (60%)
2. Smile back
3. Frown/scowl
4. Ask me about my camera
5. Apologise for being "in the way"
6. Annoyed enough to confront me in some way

When you are confronted (it happens), remain happy. Remember that you have a right to be doing what you're doing, and also remember that they have a right to be annoyed. Be respectful and listen. Do not argue or contradict, instead attempt to calm and disarm. If they're yelling at you, often the best thing to do is to walk away. How to deal with all the different possibilities is a whole article in itself, but the thing to remember here is that this is actually quite rare; I've had conversations like this maybe a half-dozen times in total, over about five years of actively shooting street photography.

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Other articles and links (please note the date of addition so people can skip to the bits they may not have read yet):

• (Aug 29 2010) - The excellent Digital Photography School had an article on this which is both interesting and decorated with lots of lovely street scenes: http://digital-photography-school.com/10-tips-for-the-aspiring-street-photographer (copy of my answer)
• (Aug 25 2010) - Alex Garcia, a photojournalist with the Chicago Tribune, has a short article on How To Take Candid Pictures of Strangers from a working photojournalist's point of view, which requires him to identify anyone he photographs, and often does involve asking first.
• A series of 11 articles on "Ways of Working" again courtesy of street-oriented blog 2point8. Very thorough, very good.
• A brief but excellent description on Flickr. Very much of the "be obvious" school of thought, though in a way that's geared to not attract attention. It could easily be describing Mermelstein in the video above. The gist is to appear as if you're already taking the picture when the person enters the frame. Very worth a read.