The Screwdriver is a tool needed for the end-game quests and to complete items for the Mysterious Engine.
You can obtain it by using a Perfect Fish Pole at Boar Island, the island west of the main island, when it's raining and at night. The Fishing bar will change to have a Green center instead of Red. You need at least 2 Green stars to obtain the Screwdriver.
The Screwdriver can be used on the following items (in progress)