I know you asked for a single part answers but since the parts work together and one choice influences another, I don't really see how that can be accomplished in a useful manner.
So I'm going to outline it all.
I'll start with the items that you need for any build.
Frame and fork.
Preferably one with horizontal dropouts - more on that later.
Headset to attach the frame and fork.
Stem - note:
the fork, stem, and the headset work together.
They can be threaded or threadless.
They must match.
Handlebars Seatpost.
The seatpost must fit snugly in the seat tube.
Seatpost clamp (Maybe.
This is built into some frames.)
Seat Front wheel.
No specific requirements other than it fit into your fork without the tire rubbing.
If you're using a front brake (and you should), you'll want the size for which the fork was intended.
There are ways to make the brakes work on a wrong-sized wheel, but it's more hassle than it's worth.
Crankset Bottom bracket.
The crankset and bottom bracket must be the compatible with one another.
The bottom bracket also be the same size and have the same thread type as the bottom bracket shell on your frame.
Pedals.
As others have mentioned, you want your feet attached to the pedals.
Whether you use toe clips or clipless is totally up to your personal preference.
Chain Tires Rear Wheel.
This is where we start to get fixie specific.
The rear wheel must have a hub that is designed to accommodate a fixed gear cog and a lock ring.
Fixed gear cog Lock ring.
You need to make sure of two things: your chain line must be ramrod straight and you must have some way of making sure that there's tension on the chain.
Let's talk about tensioning the chain first.
There are three basic ways to tension a chain on a fixie.
You can use an eccentric hub, an eccentric bottom bracket, or horizontal dropouts.
Horizontal dropouts are by far the cheapest, easiest and most common.
An eccentric bottom bracket requires a frame that is designed for one and thus, is less common and more expensive.
An eccentric hub works just fine if you have to use a frame with vertical dropouts, but it's easy enough (and significantly cheaper) to find an older frame with horizontal semi-horizontal dropouts.
Now, for making sure that chainline is stright.
The only way I know of to do it is trial and error.
Build the bike up, and then sight along the chainline.
You'll be able to see whether or not it's straight much the same way you can sight along a pool cue and see a curve in it.
If it's not, you can sometimes move the front chain ring from the inside to the outside of the spider.
Spacers on the rear hub may get you somewhere too, but you can't go too far with those.
If none of that works, you'll have to try a bottom bracket with a different spindle length.
Optionally: a front brake is highly recommended.
In some places, at least one brake is required by law.
If you have rear facing dropouts, a bmx style chain tensioner is really nice to have.
bicycles.stackexchange.com/users/4239/jimchristie
I know that some will disagree , but I would advocate using cleats .
One of the major reasons for riding a fixed is to feel your connection to the effort required to move you - and introducing a disconnect in your foot/pedal takes away from that.
Additionally, braking with your legs is easier if you're not worrying about your foot coming away.
bicycles.stackexchange.com/users/425/Unsliced
A nice old road frame (80s-vintage w/ horizontal dropout) will get you going in fine bike messenger style.
bicycles.stackexchange.com/users/None/
The same as a normal bike, minus gear derailleurs (brakes can be handy, although not completely necessary).
Get a flipflop hub, it enables you to freewheel by switching the wheel around.
Btw, you expect people to come here and list every single part for a bicycle, tis a bit weird if you ask me ...
bicycles.stackexchange.com/users/484/NimChimpsky
A frame with horizontal dropouts is preferable.
If you do end up with a frame that has Vertical Dropouts you might need to add a Chain Tensioner.
For anyone not sure what dropouts are see this pic: http://www.howtobuildafixie.com/lesson-1-dropouts/
bicycles.stackexchange.com/users/5205/Nic
The most fixie-specific bits are the hub, cog, and lock ring.
I know it's supposed to be one part per post, but these really go together.
You can't use a regular-old free-wheel hub for your fixie (at least without some modifications ).
You'll need a fixed-specific hub which has threads for a lock ring, and a cog to match.
The lock ring is what keeps the cog from loosening when you use the drive train to stop.
Surly offers each of these parts .
bicycles.stackexchange.com/users/50/Jack M.
A rear wheel is a good start for a fixie: the front is the same as any other bike, but the rear needs to be built especially for fixed-gear riding.
It will have space for only one cog on each side, typically one for a freewheel cog and one for a fixed-gear cog.
The free side will only have one set of threads, while the fixed side will have two sets of threads.
One of those is for the cog itself, and one is for the lockring used to secure the cog.
The cog needs to be secured because when you pedal backward you loosen the cog, but the lockring is threaded in reverse and will tighten, keeping the cog on.
Lots of fixie riders consider the building of the rear wheel to be part of the process of building a fixed gear.
It's the easiest wheel to build, but they are available for purchase, and usually for less money than buying the components to build your own wheel.
bicycles.stackexchange.com/users/603/Fixed God
Your hub choice depends on whether your frame has vertical or horizontal dropouts.
I converted my old mountain bike (with vertical dropouts) into a single speed building a wheel I built myself using an eccentric Eno hub.
I happen to have the wheel flipped such that it's a free wheel, but I could flip it and use it as a fixie.
bicycles.stackexchange.com/users/174/Trey Jackson
I've always wanted to build my own bike and I figured that a fixie has the least number of parts.  Please post one part per answer. Please explain what I should be looking for in the part. Also please recommend a make/model and give approximate cost.
What are the parts needed to build your own fixie?
bicycles.stackexchange.com
<fixed-gear><build><parts>
bicycles.stackexchange.com/questions/1275
bicycles.stackexchange.com/users/105/milesmeow
To build a fixie you needd the same parts as for a normal bike, minus gear derailleurs (brakes can be handy, although not completely necessary). More detailed suggestions include an old road frame and a rear wheel built specially for fixed-gear riding.  The most fixie-specific parts are the hub, cog and lock ring. Νecessary parts to build a fixie include a frame and fork, a headset to attach the former, a seat front wheel, a crankset bottom bracket, pedals, chain tires, a rear wheel, a fixed gear cog and a lock ring. Some people suggest adding a front break, and using cleats and a flipflop hub. You might also consider selecting an old road frame, which is a stylistic choice, and a frame with horizontal dropouts. In general, gear derailleurs are the only extra item included in regular bicycle parts. You could differentiate a fixie further by adding a rear wheel, using horizontal dropouts, or selecting other eccentric parts.
Νecessary parts to build a fixie include a frame and fork, a headset to attach the former, a seat front wheel, a crankset bottom bracket, pedals, chain tires, a rear wheel, a fixed gear cog and a lock ring. Some people suggest adding a front break, and using cleats and a flipflop hub. You might consider selecting an old road frame, which is a stylistic choice, and a frame with horizontal dropouts. In general, gear derailleurs are the only extra item included in regular bicycle parts. You could differentiate a fixie further by adding a rear wheel, using horizontal dropouts, or selecting other eccentric parts.