Here is what you will need to build your own fixed gear bicycle:

• A frame with horizontal dropouts (there is hope for frames with vertical dropouts...).
• A rear wheel with a threaded hub.
• A single speed cog.

Some pictures of fixed gear bicycles.
Some pictures of Martin's bicycle.

The rest is quite simple. Here's how to do it:

• Remove the shift levers, derailleurs and associated cables & housing.
• Remove the freewheel from the rear wheel.
• Install the track cog on the rear wheel.
• To size the chain, clamp the rear wheel all the way towards the front of the dropouts.
• Size the chain so that it's as short as possible and can still wrap both chainring and cog without binding.
• Slide the rear wheel back in the dropouts until the chain is the correct tension, clamp it and you're done.
• Don't forget to check your chain-line. From the back of the bike, sight along the chain and make sure it is fairly straight. If it's not, move the chainring to the inner/outer position on the crank or stick a spacer behind your cog or re-space and re-dish your rear wheel. If your chain-line is really bad, the chain will fly off when you push hard and you will fall down and go boom. Ouch...

Here is an article from the Surly web site (which I have freely plagiarized here) that is really quite useful. It explains a lot about single speed drivetrains and contains some very important cautions/considerations that the single speed novice might not be aware of.

ASIDE - What is the correct chain tension? The chain should be snug without binding at any point during the pedal rotation. Clamp the wheel in place and pedal the bike while running your fingers along the chain. You will feel the chain get tight, then loose, then tight and then loose again. That's because of things like the threads on the hub not being perfectly centered, etc. With a derailleur, this doesn't matter because the cage takes up the slack in the chain. With a fixed gear it's crucial to get it right. If the chain is too loose, it may fly off while you bounce around on your bike on a bumpy road or while spinning at really high cadences. If it's too tight, it will bind with each pedal stroke and slow you down. If you have to, err on the side of having it too loose. If it flies off, it's not the end of the world. You can just stop, put it back on and continue your ride.

If you want to get fancy, here are some more things you can do to make your fixed gear bicycle extra cool:

• Remove the outer chainring. This will require that you get some 'single' chainring bolts (which are just shorter than 'double' bolts).
• Since your chain line is probably not perfect, it really is nice to re-space the rear hub in order to line up the cog with the chainring and then re-dish the rear wheel. Go on, it's not that hard or expensive, and it will give you a stronger rear wheel and a drivetrain that will last to boot.
• Gee, did I say 'things'...? Well that's it. There isn't anything else to do. That's the beauty of a fixed gear, it's so damn simple!
• Since this bike will get used (by most of us) primarily in the winter, I might suggest getting some fenders as well. Shoot, the bike is so light already, fenders will hardly get noticed.

NOTE - Want to convert your existing geared bike to a single speed and not a fixed gear? I just found this converter at Nashbar.com.

BRAKES
Q: "Martin, I've seen movies of NY messengers where they don't have any brakes on their fixed gear bicycles, can I take those off too...?"

A: NO! Leave both of your brakes on and make sure they are well adjusted. A fixed gear bike is not a 'track' bike in that there is usually not a reverse-thread lockring holding your rear cog onto the hub. In the event you have a real panic stop, it's entirely possible that your cog will spin right off your wheel and without brakes, you'd be helpless.

MORE ABOUT BRAKES
Q: "But Martin, I've seen messengers in Seattle ride around with only the front brake, surely that must be OK...?"

A: NO! See above tip. If you ever have to do a panic stop with only your front brake and don't have mad skillz, you're guaranteed to endo. That, or your front wheel will simply push out as it looses traction and you'll fall on your side. Think of all the embarrassment you will save yourself by just keeping that second brake ON YOUR BIKE.