This light bulb is called a HYLO bulb. It has two carbon filaments that can be switched on and off by pulling a string built into its base, which moves a switch inside. The small filament is a one loop, 1 candlepower filament, while the large one is a 3 loop, 16 candlepower filament. This seems to have been a popular product at the time: You could use the large filament for normal lighting and the smaller filament as a night light for example, using the same light bulb. These were also heavily touted for "the money you'll save on your electrical bill". It seems that my bulb was almost always operated with the smaller filament, because the large filament is intact and shiny (typical of a new filament) and the little one is blackened and split in two. Also, more than half of the string is gone. There used to be another portion of string coming out of the opposite side, with a little black ball at the end of it, much like the little gray ball seen at the end of the existing string.
In the beginning, these "power saving" carbon bulbs had no strings. The switch was a simple metal plate that would press down onto another metal contact to change to the other filament by screwing or unscrewing the bulb a bit. But, you had to physically touch the bulb to unscrew it, and by then it was likely already very hot. Bulbs with that kind of mechanism are extremely hard to find. I do have one such bulb. This mechanism was replaced by the string type.
There is a very similar bulb, called the "Economical" light bulb, which was made by the same company (Economical Electric Lamp Works, N.Y.). The difference between the HYLO and the Economical bulbs is that the latter had an external switch, while the former had an internal switch. It appears that Economical light bulbs were renamed to "HYLO" bulbs when the Economical Electric Lamp company was bought by the General Electric Company. It became a division of the latter, as can be seen at the bottom of this image.
This is the Carbon HYLO bulb, but uncoiled tungsten cage filament and even coiled tungsten filament HYLO bulbs were also made. At the time of this writing I still don't have a coiled tungsten filament version, as they are extremely rare.