When Teddy Roosevelt says, "Speak softly and carry a big stick", what exactly does he mean by "big stick"...power? navy...? ...?

Kind of. Look up Big Stick Diplomacy, and depending on why you need to know this, you may want to know that it is connected to Taft's Dollar Diplomacy.

Review the Roosevelt Corollary to the Monroe Doctrine - - - it's the perfect example.

I looked up Big Stick Diplomacy. You are both right and now I get it. Here is what I found:

Big Stick Diplomacy, a nickname coined by Roosevelt in quoting the old African proverb “Speak softly and carry a big stick, and you will go far,” was the foreign policy that was later called the Roosevelt Corollary. It is an addition to the Monroe Doctrine of 1823. The Roosevelt Corollary stated that the United States had the right to enforce an “international police power” over the Western Hemisphere. It affirmed that other countries did not have the authority to cause unrest in the Western Hemisphere, most specifically in reference to conflicts between Europe and Latin America in the early 1900’s.
Eventually, the phrase “Big Stick” was used in reference to any foreign policy that made negotiations with diplomatic grace, backed by the possible threat of military force.

Source: http://www.mtholyoke.edu/~jlgarner/classwe...s/overview.html