Act functions as both a verb and a noun:
The government must act quickly to avoid worse consequences. (verb)
Polluting the stream with transmission fluid was a criminal act. (noun)
As a noun, act has several specialized meanings:
Paying for the child’s piano lessons was an act of kindness.
division of a play
Judy’s big speaking part came in the third act.
a piece of entertainment
My friend’s grandson performed a juggling act on the Jay Leno Show.
The boss was deceived, but the rest of us knew he was putting on an act.
an account of the life and deeds of a person or persons (usually plural)
Our preacher’s favorite book of the Bible is the Acts of the Apostles.
a degree or statute passed by a legislative body
The Volstead Act prohibited intoxicating beverages and regulated the manufacture, production, use and sale of high-proof spirits for purposes other than drinking.
Action is an abstract noun formed from the verb to act. It also has more than one meaning:
She has retained counsel, David Shiller, to bring an action against The Toronto Star.
Captain Martin, although badly wounded, survived the action.
In teaching the Itsy-Bitsy-Spider song to a child, first teach the words, then the actions.
event or series of events in a story
The action in the film Gigi takes place in turn-of-the-twentieth century Paris.
Using action and act interchangeably is appropriate only when the sense is “deed” or “thing done.” Action has meanings that do not apply to act, as in the following statements:
There can be no excuse for his actions on the day of the disaster.
He lived a life of action.
At the time, the action of asbestos on the lungs was not widely known.