Often times, musicians are asked to "improvise" their part from Chord Symbols. This is a very common task. Perhaps you will be making an arrangement of a piece of music, and all you have to start with is a lead sheet (showing the melody) and Chord Symbols. The Chord Symbol provides all the information about the "harmony" of the music, and are quite easy to write. Even when making arrangements, you might want to give the rhythm section (guitar, bass, keyboard, drums) a simple chord chart, and/or you might supply Chord Symbols to other instruments for "solos" or "improvisation" (don't forget to transpose your Chord Symbols when an instrument is a transposing instrument)..
Components of a Chord Symbol: First, we show the ROOT of the chord. Second, the chord quality (is it Major, Minor, Diminished, etc.), next, extensions if any, and last, if it is inverted (not shown in diagram below).
Inverted Chord Symbol: If we take the example above and write it this way CMaj9(#11)/E, we have a first inversion. We are telling the players to play this chord with the Major 3rd ("E" in this case) on the bottom. If we want the 5th or second inversion on the bottom, we would write this: CMaj9(#11)/G .Chord Symbols can be as simple as a basic three (3) note triad, or can be complex chord structures like CminMaj9(#11) .
Polychord Symbols: Polychords can also be written using Chord Symbols, like Dmin/Cmin (tells us to play a Dmin in the upper register and a Cmin in the lower). Chord Symbols can also indicate special chords like rock power chords such as C5 (tells us to play the root and the 5th only), or can also be stated as C(omit3).
Now that you have an understanding of Chord Symbols, please see the page about Chords which shows many more examples of how to write Chord Symbols and the notes which make up the chords.