Now that I've stirred up some dissent with my views on bar width and height, it's time to ramble on about the mechanics of handlebars.
Stem Clamp Size: This is easy; most modern bars have a clamp area of either 25.4mm or 26mm. If you want to use a 25.4mm bar in a 26mm stem you need a shim, like this lovely Nitto product. The other way round won't work, unless you can find one of the rare negative shims on E-bay.
Most older French bars and stems use a 25.0mm clamps. I've used these bars in 25.4mm stems, but it's best to make a little shim from some beer can aluminum. (BTW, obtaining shim stock is the only acceptable reason to drink beer from cans. I seem to need lots of shims.)
Older Cinelli bars and stems are 26.4mm and there are a few other odd sizes. So it's wise to check when buying older European bars.
Stems and Forks: Again this is simple. Most 1-inch forks use 22.2mm stems. Older French forks use 22.0mm stems. Fortunately these are close enough that a 22.2mm stem will sometimes fit in an old French fork. And if it doesn't, 10 minutes of sanding (on the stem not the fork) will allow it to slide right in.
Inexpensive older American and Japanese frames sometimes used a smaller diameter, 21.15mm, stem that is becoming very hard to find. This size is also common on BMX bikes.
There are also some newfangled 1-1/8" forks and stems, but it's too soon to say if they'll catch on. And some tandems use 1-1/4" forks/stems.
Bar Diameter: The outside diameter (OD) of bar tubing varies a bit by manufacturer, but the standard for road bars is 23.8mm and MTB and city bike bars are around 22.2mm. That means that MTB brake levers and shifters will not fit on road bars.
The inside diameter (ID) of road bars is around 20mm and for MBT bars it's about 17mm. This varies with the exact type of tubing used, but inverse brakes and bar end shifters generally will not fit MTB/city bike bars. They commonly require an ID of not less that 19.6mm.
One way around this problem of compatibility is to buy old French city bike bars which were often made with road bar diameter tubing. With these you can use bar ends, inverse shifters, and some of the old-style brake levers with band-type clamps.
A way to make everything fit is to use chrome molly, rather than aluminum, bars. The steel is stronger than alloy, so it can be thinner, resulting in bars with the OD of MTB bars and the ID of road bars.
As with most thing, there are some odd, mostly French, bar diameters out there.
Digital Calipers: Cheap digital calipers are available for about $15-$20. If you work on older bikes they will save you countless headaches. And you don't even have to learn to read a vernier scale. Try Harbor Freight or E-bay. Maybe we should stock them?
What have I left out?