What are the parts needed to build your own fixie?
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.