A double sharp is an accidental that is used to signal double sharpening of a note. Double sharps are most likely to occur in keys containing (lots of) sharp already. If an F in the key of C is sharpened, it will become a F#. If that piece is transposed to C#, the F# will become a F## and shown with a double sharp.
A F## will sometimes be enharmonically exchanged to a G-natural to accommodate reading.