The big stick, in the phrase "Speak softly and carry a big stick" refers to the ability to use violence or force if necessary. The phrase is supposed to encourage people to avoid aggression by using caution but also to have a weapon at the ready.
The phrase was popularized by Theodore Roosevelt who wrote it in a letter to Henry S. Prague on January 26, 1900. Roosevelt states that the origins of the phrase stem from West Africa, although that has not been confirmed. The full quote from Roosevelt's letter states, "Speak softly and carry a big stick; you will go far."