Curb chain resembles cable chain that has been twisted and flattened, so that all the links lie flat. Curb chain's flattened loops are typically oval and all the same size. In some chains, the links in the chain may be made round, and then fed through a rolling mill for greater flatness. So you may see chain that has totally round wire, or it may have flattened spots.
Curb chain and gourmette chain are the same style of chain: flattened ovals. And we owe this chain to horses!
When we hear "curb" today, we probably think of the corner between a sidewalk and the road. Yet that isn't what that word meant until about 1791. Before that, "curb" referred to the strap passing under the jaw of a horse.
Let's take a look at "gourmette," another name for curb chain. This is a French word that gave us the word "grommet," meaning a ring or wreath of rope. Gourmette chain was originally - and still is - used to link a horse's bit, under his chin. You can imagine the horse greatly preferring a chain that lay flat against his jaw! While many curb straps are leather, many bits still use metal curb chain today.
There also seems to be a connection between coats of mail and curb chain. "Panzerkette" means curb chain in German, and "panzerhemd" means, literally "armor shirt." While my research didn't uncover any German chain maille armor made in a curb chain pattern, it is certainly possible - and perhaps that inspired the style of horse curb chain. (One source insisted "curb chain" is named after "panzerhemd" - but the horse connection seems much more likely!)
Pictures of Curb Chain