Runes are basically magical tools that you can use via your Sheikah Slate to perform certain functions. These include solving puzzles found inside Shrines, finding hidden chests, and moving heavy objects. Listed below are the four main types in the game along with what they can be used for.
Magnesis Rune
Location: Found inside Oman Au Shrine of Trials
This Rune allows you to lift and move metallic objects such as metallic chests out of the water.
Cryonis Rune
Location: Found inside Keh Namut Shrine of Trials
This Rune allows you to build ice pillars from water on any water surface which you can then climb to access otherwise inaccessible areas. These Runes can also be used to solve puzzles.
Stasis Rune
Location: Found inside Owa Daim Shrine of Trials
This Rune allows you to stop an object while it retains its momentum. Upon release, the object can be moved which is otherwise impossible. You can use it to move large boulders to reveal hidden chests and solve puzzles.
Remote Bomb
Location: Found inside Ja Baij Shrine of Trials
The bombs come in two forms, cubic and sphere. These bombs can be used to deal damage to monsters and destroy weak objects and walls. They are detonated on command with the press of a button.