Palmolive is a brand of dish washing/washing up liquid used primarily for cooking and eating implements. Palmolive 'soap' first came about in the early 1900's when the B.J. Johnson Company created a soap made entirely out of palm oil and olive oil, hence Palmolive. It turned out to be so successful that the company was soon after renamed 'Palmolive', after their flagship product. Palmolive later merged with the Peet company to become Palmolive-Peet and, in 1928, that company merged with Colgate to create the Colgate-Palmolive-Peet Company. Peet was dropped, leaving Colgate-Palmolive as the company that brings us our Palmolive washing liquid.
It should be said that the formula has changed much over the years to reflect the science of the day, and Palmolive now contains neither Palm Oil or Olive Oil.