ohashi is correct.
I would add that what you as a player do after you drop a broken stick depends upon where you are and in what direction you are skating on the ice, where the puck is, and which team has possession of the puck.
If you are in the offensive zone and your stick breaks, you drop it out of the way of the play so your team's offense is not disrupted. If your team has the puck, you retreat to your bench where an equipment manager hands you your replacement stick which he got because he saw you broke yours a few seconds ago.
If your team lost the puck in the offensive zone as a result of a shot where your stick broke, you drop your stick in a way most likely to disrupt the other team, and you start playing defense without a stick until you can get back to the bench for a new stick, or until a player who can more easily get to the bench hands his to you.
If your stick breaks in the defensive zone, you are more likely to just start playing defense without a stick.
If a goaltender loses or breaks his stick, everyone in organized hockey has been taught to offer their stick to the goalie. Some goalies want any stick. Other goalies will play without a stick until his/her stick can be retrieved or until a stoppage of play.
In most sets of hockey rules, the goaltender is the only player who is allowed to continue playing with a broken stick.