Disclaimer: I am not a
lawyer, a
NASCAR driver or an automotive
engineer. I take no responsibility for the results if you follow these
directions and they don't work or mess your
situation up even more. That said, this is all based on actual
experience, not just random
extrapolation.
So there you are, driving along, you go to stop for a red light up ahead, and that always-firm, ever-reliable brake pedal suddenly feels like it's mounted to a bowl of warm tapioca. The first thing to remember is: DON'T PANIC. If you panic, you'll probably just shove the pedal to the floor and start hyperventilating. This is the absolute worst thing you can possibly do.
The second thing to remember is that you probably still have some stopping power. Maybe even quite a lot, if you can figure out where it is. Pump the brakes. If that works, you can just possibly get out of this with nothing more than a little underarm sweat and maybe a scuffed tire or two. Even if it doesn't, there are a still a couple of tricks up your sleeve.
Shift into neutral. This is safe to do even without stopping first on every automatic I've driven (unlike trying to shift straight into Park, which most transmissions just won't do at speed), and on manuals you don't use the brakes while shifting anyway. On a level road you'll notice your speed dropping immediately, maybe faster than you anticipated. Keep pumping your brakes after you do this -- both for whatever power they still have, and to be sure and warn the people behind you that you're stopping. If it's safe to do so, you may want to shift into a low-drive gear first -- the engine will brake you faster than tire friction. Be careful though -- many automatics won't shift to the lowest gear till you've dropped below 40 MPH or so, and if you find yourself rolling downhill and accelerating with the transmission fighting it, you can cause damage (especially if you roll far enough to exceed the lowest-gear-safe speed). At best you risk stalling the car and losing your power-steering assist when the engine cuts out, so be careful; if you have the level-ground distance to spare I advocate just going to neutral. drinkypoo advises that most manuals don't even have a synchro in first gear because it's so easy to blow up the transmission shifting straight into first. If that looks like Vulgate Sanskrit to you, read up on transmissions here:
Manual transmissions
Automatic transmissions
Ever see skiers on a hill? They don't just go straight downhill -- that's a good way to accelerate to a massively unstable speed, crash into an obstacle and die. Skiers go side-to-side, and so (assuming the road isn't slick or otherwise unsafe to do this on) should you. Wiggle the wheel from side to side a bit. This does two things: increases the distance you travel in a given amount of lane, and converts some of your car's energy to lateral motion instead of forward. Be very, very careful doing this -- in fact I hesitate to include it here, because it can go horribly wrong if you do it on a wet or icy road -- and think hard before you do it in the first place.
So by this point, one way or another, you're at least slowing down. Now take a second or three to look ahead of you. Are you still likely to hit cars up ahead? If so, flip on your hazard lights, tap your horn a couple of times, and maybe flash your headlights a time or two to get people's attention.
The last few miles per hour can be the toughest to shed. If you're going uphill, you're probably OK -- in fact you may have on your hands the new problem of how to avoid rolling backwards after coming to a stop. It's not good for your transmission, but you can shift back into gear and out of it in cycles until you can get into a stable point where you don't need the brakes to stay stopped (if your parking brake still works, this is the time to use it -- it's useless at speed and you'll just burn it out if you try to stop with it right off). (Some cars don't have a separate parking brake, the lever merely engages one or more of the existing brake assemblies, therefore if you have no regular brakes, you may have no parking brake either. So don't depend on this to work, but it's always worth a try.) If you're in the lowest drive gear, this may be the point where you finally shift out of it into neutral.
If you're going downhill, or you're on level ground and still going too fast, this is the time to look for a safe crash site. Rubbing curbs will often bring you to a stop (while not being great for your tires, but that, at this point, is a lesser issue). Try to avoid utility poles, in favor of larger, denser things like Jersey barriers that you can sideswipe. Use shoulder lanes and empty travel lanes for additional distance if they're there; it's part of what shoulder lanes are for. Try to get over to the right if you can -- you have a better chance of sliding by to the right and making a turn (relatively) safely than going straight through or making a left.
If worse comes to worst, and you just can't avoid hitting other cars, try to at least do it at an angle, not straight-on. A straight-on rear-end crash can push them into the cars in front, up to 3 or 4 cars ahead in line.
In the aftermath, try to remember what you observed leading up to the point where you lost the brakes. Was there smoke? Could be a busted seal -- that smoke might be leaking brake fluid vaporizing between pad and rotor. Did you lose the brakes all at once or gradually over a couple of stops? Was there a squealing or grinding noise? A funny smell? Your car is going into the shop after this, and all these things can help the mechanic figure out what went wrong, if it's not obvious, and check to see if you're in danger of it going wrong somewhere else.
Lastly, like any good information provider, I have to end with the appeal for help. If you have corrections or additions to any of the above, feel free to send them to me (or, if you're a noder, just contribute them as writeups of your own).