What things should I look for when purchasing a fixed gear bike? What features are important or of note? What are appropriate prices to pay? Are there specific brands which provide a clear advantage over others?
You are likely going to get some opinionated comments and answers in response to this question.  There are lots of good brands (frames and complete bikes) from all over the world.  There are more and more options showing up in local bike stores, so if you can it will be best to go have a look.  Even if you choose to purchase online, having spent some time looking at physical hardware will help you make your decision.

I'm going to make an assumption that you're asking about a fixed gear bicycle for road use as opposed to a track.

If you haven't read Sheldon Brown's detailed article then please do, it's a great place to start.  Without reiterating all of the detail in that article I'll simply point out a couple of highlights:


The number one thing that you should look for is a brake.  Again, others may offer dissenting opinions on this, but riding a fixed gear bicycle on the road without a brake is extremely dangerous.
Chain tension mechanism.  You will have to be able to tension the chain on your fixed gear bike as it lacks the derailleur and pulley mechanism that handles that task on traditional geared bikes.  This will likely be accomplished by means of horizontal rear wheel dropouts within which the wheel moves back and forth for adjustment.  You may also encounter models with an eccentric bottom bracket mechanism.  These typically add some weight and may be of less interest to you unless you happen to go for a rear disk brake (in which case moving the rear wheel to adjust chain tension is not an option)
If you are new to the world of fixed gears then you may be interested in having an option to run a freewheel as well.  A flip-flop hub can offer the best of both worlds in that you can ride fixed or flip the wheel around and ride with a freewheel.  This adds a small weight penalty and absolutely requires that you install a brake of course.


As for price, brands, geometries, gear ratios, frame materials etc. this is very much personal preference and the rules are the same as for any bicycle.  You can typically splurge a little on a fixie because there are far fewer other components that you have to purchase and maintain.