Tom,

That style of watch chain is named "Albert" after Prince Albert, son of Queen Victoria. The tapered link design was favored by the prince. They come in double Albert and single Albert versions.

The double is designed for use with a vest, it has a T-bar in the center, with a chain and swivel hook on each side. The single just has a T-bar on one end and a swivel hook on the other. Either may also have a fob drop.