From the Super Mario Wiki
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Skull Switches are items found in New Super Mario Bros., which cause the defeat of the boss they are in the room of. They replace the axe, found in the original Super Mario Bros., and function the same way.
In total, Skull Switches are found in three boss fights; the first with Bowser, the second with Dry Bowser, and the third with Bowser and Bowser Jr. In the aforementioned boss battles, Mario's goal is to avoid the attacks of the bosses and hit the Skull Switch, causing the bridge which Bowser is standing on to collapse and send him plummeting downwards to whatever may lie beneath. When Mario sets foot on the safe platform with the switch, Bowser/Dry Bowser will run towards him and try to toss him back onto the bridge. Even though Mario can defeat Bowser with fireballs, he still needs to press the switch as in the original Super Mario Bros.