The word play can have several meanings.
Play can mean ‘be cooperative’.
She needs a loan, but the banks won’t play.
If something comes into play, it becomes active or effective.
Luck came into play.
Make great play of
To make great play of something is to draw attention to it in an exaggerated way.
Play about / Play around
To play about is to behave in an irresponsible way.
Stop playing about. You are a grown up boy now.
To play along is to pretend to cooperate.
Play it by ear
To play it by ear is to go about doing something without a plan.
Play something down
To play something down is to disguise the importance of something.
Play into someone’s hands
To play into someone’s hands is to do something that would give them an advantage without meaning to do so.
Play someone off against another
Play someone off against another is to create conflict between two people for your own benefit.
To play on something is to take advantage of someone’s weak point.
He played on her emotions.
To play up is to cause problems.
Play with fire
To play with fire is to take foolish risks.