The first, actual “curling iron” was patented by Sir Hiram Maxim (the same fellow who invented the machine gun) in 1866, but the device itself dates back over 6000 years. Considering antique curling tongs are seen or referred to in numerous cultures, it is doubtful a single person is responsible for the invention. Beauty is certainly not exclusive to one country or culture!
Typically, curling irons (or curling tongs) were made of iron, bronze, and other metals. Heating was done by the straightforward method of inserting into the fire or laying atop a hot stove. The handles were usually made of a different, non-heat conducting material such as wood, bone, ivory, etc. Most handles were plain and utilitarian, but some expensive models had nickel-plated handles carved ornately. Note the Victorian Era example in the lower left corner of the image below. The tong length and diameter varied, depending upon the intended use, which was not only for a woman’s hair but also for wigs and beards.