Big Stick Diplomacy refers to President Theodore Roosevelt's foreign policy.
'Speak softly and carry a big stick, you will go far' - it was a combination of intelligence and decisive action while dealing with foreign issues. It meant having strength when things go wrong but trying to negotiating peacefully.
This foreign policy was applied during U.S. intervention in Latin America, for example in the Venezuelan Affair in 1902.
It was also used during the 'Canal Diplomacy' while dealing with incidents from Nicaragua and Panama.
Another example is Roosevelt's involvement in the Russian-Japanese War Peace Treaty in September 1905. He negotiated the treaty between these two nations 'speaking softly', which brought him a Nobel Piece Prize the following year.