According to what I can find, this is supposed to cancel a double-sharp and give a single sharp on a note.
But I ran into it where there is no double-sharp to be cancelled. There's a G flat in the key signature, however, and the idea appears to be to cancel the flat and make it G-sharp. Why it couldn't be written as an A-flat I don't know.
The accidentals indicated are always absolute and not relative and they are unrelated to the key signature. The Natural sign cancel all previous accidentals, ie you get a G (white key), irrespective of what is in the key signature or previous accidentals. There is a common misconception that the natural sign cancels only one sharp, but that is untrue. In some editions, you will find a double natural sign after a double sharp or flat, but that is strictly not necessary, the single natural is enough.
If you want to get a G sharp after a previous double sharp, you need to write one sharp (assuming the key signature does not have one, and we are in the same bar). Most editors for clarity will put a natural sign and a sharp by courtesy.
In your case, if there is a G flat in the key signature, and you see a natural G, then it is exactly that, a natural G (ie white key). If the author wanted a G sharp, he would put a sharp or possibly and by courtesy a natural and sharp.