Countries that currently, or in recent history have depended upon the United States. Often used in an arrogant manner by some Americans who think that somehow showing support towards a nation should somehow give them a stake in that nation. Also often used by people noting that the culture of a country seems, or that a country shows a lot of support towards the US, or because of their cultural similarity to the US.
First usage: The United Kingdom due to the assistance from Americans in World War II, is sometimes a considered a 51st state.
Second usage: The United Kingdom is considered a 51st state due to their government's support of America in the Iraq war.