What features are important or of note? What are appropriate prices to pay? Are there specific brands which provide a clear advantage over others?
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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:
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.
I would break it down into the following categories...
Here's the difference between a track bike and a road bike in terms of geometry. What this translates to in terms of handling will take some writing...
Track bike geom (Bianchi super pista)
Road bike geom (Bianchi Vigorelli)
From my riding style and my delightful experience with my converted old steel 10 speed, what I like most in my bike is:
Now the explanation: I use this bike for training and fair-weather commuting. I prefer it over my heavier bike (with fenders, lights, internal-geared-hub, rack) because it is so much faster (for being lighter), efficient (for having a simpler drivetrain and skinny tires) and nimble (for having a narrow handlebar and a trusty front brake). Being all-steel drivetrain (chainring, chain and sprocket), the chain oil keeps clean for much longer than with aluminum chainring, which makes the oil black too soon (and it stains the clothes a lot more!).
What I don't like in this bike:
Well, I talked a lot, but I avoided to mention brands, I prefer to mention features, many good and pricey bikes might not have your desired features, and many apparently crappy bikes might have just what you need.
I bought one used and immediately spent 20% of paid price to replace some parts (crankset because of quality, stem and handlebars because of fit). That's the good part of fixies: you never need to spend a lot, since there is not so many parts to swap.
Hope it helps!