Fetching details from the dictionary...

to put; to place
to leave (behind)
to establish (an organization, a facility, a position, etc.); to set up
to appoint (someone to a certain position); to hire; to employ
to place (one's trust, one's faith, etc.); to bear (in mind, etc.)
to put down a tool (e.g. a pen) hence stopping what one is doing with that tool
to take in (boarders, etc.); to provide lodging in one's house
to separate spatially or temporally
to do something in advance
to leave something in a certain state; to keep something in a certain state
机の上に本を置いたよ。
I put a book on the desk.
傘を電車に置いてきた。
I left my umbrella behind on the train.
ここに荷物を置いてもいい?
Can I put my luggage here?