'Cow Power Program' was started in vermont, the United State.

Question

(In this sentence, why is the quotation mark was used? When do we use the quotation mark?)

Answer 1

Quotation marks should not be used here.

This webpage explains when quotation marks should and should not be used:
http://grammar.ccc.commnet.edu/grammar/marks/quotation.htm

The end of that sentence should be "... in Vermont, USA." (Or you could leave out USA if you're writing in the US.)