Before the curling iron, women used to put a towel over their hair and use a regular iron on it to get it straight. Then came hot rollers. This was the precurser to to curling iron. The crimping iron then came out in the 80's.
The person credited with inventing the curling iron is Charles L. Nessler, in 1906. His process was extremely complicated and expensive, and could take up to 12 hours.