I've recently started riding a fixed-gear bicycle and I love it. It's simple, light and fast. However, being new to "fixies" I have two brakes with a drop-down bar, like a road-bike. It just feels safer. Many fixies I've seen have one front brake, usually from a simple brake lever in the middle of the bar.
What if you suddenly brake hard, wouldn't the momentum flip you over the front? Is the front brake a "minimalist" look just to look different, with no regard to its possible safety downside?
It feels safer because it is safer.
With a fixie you can brake the rear wheel from the pedals. In theory, you only need a front brake. The theory is great for experienced fixie riders. For novice fixie riders a second brake on the rear wheel is wise. You have to learn to control speed with pressure the wrong way on the pedals and if you feet ever leave the pedals you can be in deep trouble with just a front brake.
In theory, even the most experienced could find themselves in a situation where a rear brake would be helpful.
You may ask why not put the one brake on the rear - this is ineffective for hard braking, as vast majority of your braking comes from the front wheel.
Longtime fixed-gear rider here (~14 years, thousands of miles), and I run a front brake only.
I won't go fully brakeless because I consider it too risky. I am a year-round commuter and I have actually had a chain *snap* twice coming down a hill in the winter (in New York state - I blame road salt for corroding the chain). If I did not have the separate front brake in those moments I would have careened down the hill unable to stop. Not good.
I don't run a rear brake for a few reasons:
From the phyics of things, braking on the rear wheel using the pedals or using a conventional brake will not make any difference. The limiting factor for braking is the traction you get between the tire and road, and the unloading of the rear wheel due to the torque of the forward momentum being reduced. Whether the braking of the wheel is done via the pedals or via an actual brake doesn't change the limiting factor.
The center of gravity of rider and bike is well above the road; during deceleration this creates a torque that has to be balanced by vertical forces on the wheels. The front wheel will be loaded (increasing its braking capacity) and the rear wheal will be unloaded. This torque is the reason that braking too hard on the front brake will flip you over the handlebars, but it is also the reason that the front brake can contribute more to the stopping force than the rear one. It doesn't have to contribute more force, you can come to a stop braking only with the rear wheel, but the front brake is more effective.
As to the reason that fixies don't use rear brakes: it's the weight! A fixie is an absolute minimum, trimmed down bicycle that eliminates everything you don't absolutely need. [Edit to address @DuncanRMI]: They were developed for track racing, and for them, low weight is high tech. (Think carbon frames, etc.) That is its appeal and that is also what makes it cool.
As to being unsafe--the chance of your brake cable snapping is probably a lot higher than the chance of the chain breaking, so braking by the pedals comes out on top on this measure.
The front brake is kept to fulfill traffic regulations which in most countries require two separate means of stopping the bicycle. But of course, as @MattMenzeski has already mentioned in his answer, you might want one yourself for safety reasons even when it's not legally required.
Because a rear brake on a fixie is not very useful. First, a fixie (unlike a single speed bike with freewheel) allows some braking by resisting the pedals. Second, braking removes weight from the rear wheel, so rear brake does not do much to slow down the bike in any case. Since most of the braking is done with the front brake anyway, you could as well leave out the rear brake and use your legs for what the rear brake can do.
If you don't know what you are doing, you can certainly flip over the bike with front brake. With rear brake you're guaranteed to not slow down quickly, no matter if you can brake or not, and you can still slide and fall.
Bicycles have one effective brake (in the front) and a backup "slow-down device" in the rear.
What if you suddenly brake hard, wouldn't the momentum flip you over the front?
Yes, of course.
If you never use the front brake out of fear to flip over the bars, then you won't be trained to use it properly. Then when a driver piles on the brakes right in front of you because something happened further ahead, and their car screeches to a halt... your hand will squeeze the rear brake because that's what you trained your reflexes to do by always using the rear brake.
About a second later, you realize you're not slowing down nearly enough... and your rear wheel is also skidding. So you try the front brake. But you've already wasted precious braking distance, and you never use the front brake, so you can't automatically apply the proper pressure by instinct. So you either don't squeeze enough and crash into the stopped car, or panic and really squeeze it, then flip over the bars and crash face first into the rear window of a stopped car.
However, if you use the front brake most of the time, especially in situations when there is no emergency and the rear brake would be fine, then you can really train your reflexes to use it properly. Then, when you actually need to brake hard to avoid crashing, you will be able to apply the maximum braking power that still keeps the rear wheel on the ground almost immediately.
It's one of these paradoxes: if you're too afraid to flip over the bars and don't learn to use the front brake, you'll eventually flip over the bars anyway.
Do not practice on a slippery surface, though.
A rear brake is not effective on any bike. A rear brake makes you skid, because when braking, weight shifts from the rear of the bike to the front, sometimes lifting the rear wheel. A front brake makes use of the larger weight on the front wheel to brake more effectively.
