What does the "white man's burden" Pear's Soap advertisment mean?
I assume that you are asking about the advertisement for Pear's Soap that features Admiral Dewey.
If so, the meaning of this ad is that white people had a duty to go out and civilize the other people of the world. Pear's is trying to use that idea (and the celebrity of Admiral Dewey due to the Spanish-American War) to sell their product. The term "white man's burden" refers to this duty. It comes from a poem by Rudyard Kipling.
This ad, then, is an example of how Americans in the early 1900s felt that they were superior to people of other races. It represents the idea that their superiority makes it right for them to go and conquer other races of people.