Planning on building a fixed-gear/single-speed from an old Schwinn World. What new parts will I need?
Fantastic ! Building a fixed gear conversion is a lot of fun.

tommy_o gives a great list of all the parts that go on a conversion, but when I try to imagine what it's like to have an old bike that you want to hack on, I think it's helpful to imagine what you'll have to change about your bike, so I'll give that a shot.

As long as the existing bike is in decent shape, you actually have a lot of freedom in deciding whether to get new parts or reuse existing ones. At a minimum, though, you'll need a new rear wheel, and then from there it gets more and more up to you.

Wheel

You can either buy a wheel, or build one yourself. I've found that buying is actually cheaper, but I think it's a good exercise to build at least one yourself to see how it's done. Either way, like tommy_o mentioned, you'll need to pay attention to the size of the rim that you want to get. If your bike has 27" wheels and you decide to get a 700c rear replacement, then you'll need to be ok with having different size wheels, or you'll also need to get a new front wheel.

If you decide to build, you'll need to buy :


hub
rim
spokes
nipples
rim tape


You'll also need a spoke wrench, but that's a good tool to have around anyway.

I highly recommend reading a copy of The Bicycle Wheel by Jobst Brandt. The obligatory Sheldon Brown link is also in this section, because his description of wheelbuilding is a great read.

Hub

For the hub, there are indeed different sizes, both in terms of axle length and flange size. However, for the axle length, there are really only a few different standard sizes, so you should be good to go with a road hub.

For the flange (which only matters if you're building a wheel), pretty much anything will do, but the flange will have a significant impact on the length of the spokes that you'll need. I always take my rim and hub to my local bike shop and get the folks there to cut spokes for me, since it seems like some sort of black magic.

Drivetrain

Once you've got a new rear wheel, you need to connect it to the pedals. At a minimum this means you need a cog on the rear wheel (typically hubs do not come with cogs), but depending on what size you get, and personal preference regarding the chainring, you might also need a new chain and chainring.

Cog

You'll definitely need a cog, the gear that goes on the rear wheel. You should also get a lockring.

Getting the cog is actually a little tricky, because it turns out there are two different widths of bicycle cogs, and they are not interchangeable. (There are actually two different widths of bicycle chain, but the chain, chainring, and cog all need to work together.) The wider setup, usually seen on track bikes, is 1/8", and the narrower one is 3/32". Here you'll need to decide if you want to stick with the 3/32" setup that's already on your bike, or move to the 1/8" ; again, mostly this has to do with personal preference, but be aware that whatever you pick will affect your options for the other items.

If you get a flip-flop hub, you should probably get a freewheel cog as well as a standard thread-on one. Typically the freewheel cog has a higher tooth count than the fixed one, since you won't need to pedal like a rabid gopher while you're coasting downhill on your freewheel side.

Chain

You'll definitely need a new chain if you go with a 1/8" drivetrain.

Sometimes it's a good idea just to get a new chain, even if you're sticking with the existing sizes. New chains are really nice.

Chainring and crank

There'll already be a crankset on your existing bike, which can probably be used just fine for a conversion (provided it works with the chain/cog). There are a couple things to think about it however :


You should make sure that one of your existing chainrings has a tooth count that you can use with your rear cog. The important thing here is the ratio between chainring and cog : a lower ratio makes acceleration easier but limits your top speed. Also if you plan on skidding, it's easier with a lower ratio.
If the chainring you want to use is on the inside of your crankset, then you might want to remove the outer chainring so your pant leg etc. won't get torn up on it.


Also, on a fixed gear, the length of your crank could be important : If you're taking a tight corner, you might lean over far enough that the bottom of the crank scrapes the road as the pedal turns, which can lead to a wreck. You can get a crankset that has shorter arms to avoid this, though if you're not into tight cornering it might not be a problem. It's something to be aware of though, since you never really know when you might need to pull an emergency maneuver.

Other stuff

Once you've gotten the rear wheel and drivetrain set up, you're technically good to go. You can opt to switch many other things on your bike, but chances are that it already has a seat, headset, fork, handlebars, etc.