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

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

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.)