Peanut butter has its origins in South America.
Although they didn't "invent" the spread that we buy in jars, the earliest historical reference to a peanut paste can be traced to the Incas and Aztecs.
As far as naming a specific innovator goes, African-American scientist and inventor George Washington Carver is often (incorrectly) credited with inventing the stuff. In reality, he impressively devised 300 recipes and uses for peanuts, which included a peanut paste, though he didn't invent peanut butter.
The first US patent for a peanut flavoring paste (aka "peanut-candy") was actually awarded to Canadian chemist Marcellus Gilmore Edson in 1884. However, two American physicians did contribute to peanut butter as we know it: John Harvey Kellogg (the man behind Kellogg's cereal) patented a method for turning raw peanuts into peanut butter, while Ambrose Straub patented a peanut butter-making machine.