I believe they were actually the National Watch Company, based in Elgin, Illinois, but incorporated the city into their official name at some point.
Elgin concentrated on middle-class watches: lots of 7 through 15-jewel jobs. Not the best, but you don't need 23 jewels to get the job done. This was my father's favorite brand, so I have a bunch of them.
This is my Elgin Grade 288, made in 1918. It is size 18, 7 jewels, pendant wind, pendant set. The case is from Star Watch Case Company, out of nickel. This is unusual among my old Elgins in that the movement is on a hinge and swings out. Most of my Elgins have a screw back.
Acquired: 2019
Circa: 1919
Current condition: not working, don't know the problem.