'1 1/8' vs '1/2 1/8' chain/chainring Some 1/8 chains and chainrings are sold as "1/2 1/8" and some as "1 1/8". What is the first number describing?

My understanding is that practically all modern chains have a 1/2 inch pitch.
There are two measurements, given as pitch x width. The pitch is the distance between rollers and width is the width that the sprockets have to fit through. The pitch is generally 1/2" on modern chains, but some old bicycles (esp. old track bicycles) use 1" pitch chains (skip link or block chains). 

The width changes depending on number of speeds (so you'll have a lower width for higher number of speeds). So single speeds will often use 1/8" (some have sprockets designed for 3/32" though, so you should check), 8 speed will use 3/32" , higher number of speeds will use 11/128" (11 speed), etc. [possibly different between manufacturers].