In music, indicates that a note is to be raised two half-steps, or a whole step, from it's printed pitch. For example, in the case of a C double-sharp (usually notated with a symbol that looks like an "x"), you go up two half steps and play the pitch otherwise known as D.
People often wonder why double sharps are neccesary, because any pitch notated as a double sharp can always be written enharmonically without the use of double sharps or double flats. The answer is in the spelling: when writing chords, one does not write only by sound, but also by how the chord is logically structured and spelled. A parallel in language is the fact that you can often spell things so that they sound the same: for example, "subtle" could be spelled "suttle" and would still sound the same, but it's not correct to spell it that way. However, where spelling is just a matter of agreed convention, the reasons for double sharps and double flats are more fundamental.
To illustrate, imagine you're writing a G Major scale: G, A, B, C, D, E, F#, G, very simple, right? Now make it a G# major scale: we have G#, A#, B#, C#, D#, E#... now what?? The next note looks like it should be a G, but we've already used G, and a scale only uses each letter name once. This is where the double sharp comes in: we end the scale Fx, G#, and everyone lives happily ever after.