Chromaticism refers to the use of pitches, chords, and keys not associated with diatonic collections. The etymology of the word chromatic, which refers to colour, gives us a clue as to its function is in nineteenth-century music: it provides inflections to diatonic harmonies.
A chromatic pitch is any note not contained within a given diatonic collection. For example, in C major, C-sharp, D-flat, D-sharp, E-flat, F-sharp, G-flat, G-sharp, A-flat, A-sharp and B-flat all represent chromatic pitches. For a chromatic pitch to function 'chromatically', however, it must resolve in a logical way to a diatonic pitch, otherwise the overload of 'colour' undermines the integrity of the key and begins to suggest a modulation to a different key or a non-diatonic modality. As a general rule, chromatically raised tones resolve upward while chromatically lowered ones resolve downward. Hence, chromatically introduced A# usually goes to B while Eb would have to go to D.
In nineteenth-century music there can be no pitches without chords, which more fully suggest harmony. In C major, chromatic chords include all those outside the diatonic framework, including C minor, C-sharp major and minor, D major, E-flat major and minor, E major, F minor, F-sharp major and minor, G minor, A-flat major and minor, A major, B-flat major and minor, and B major and minor. The way these chords are used in nineteenth-century music is not arbitrary and each chord has its own specific quality and compositional implications. Most obviously the level of diatonicism, or its displacement around the cycle of fifths, of a chromatic chord makes it sound more or less nearly-related to the tonic (that is within diatonicism, although other theories exist, notably certain neo-Riemannian theories).
Finally, keys, which may provide large-scale harmonic structure in nineteenth-century music, may also be chromatic. Given that gamut of keys for most music of the eighteenth century is diatonic - most often creating a tension between tonic and dominant (Charles Rosen, The Classical Style, 1978) - the use of chromatic keys opens up a vista of new tonal possibilities. Composers such as Beethoven and to an even greater extent Schubert, are examples of the first composers to explore this.
Historically, the prolongations of chromatic pitches, chords, and keys, increasingly undermining a clear diatonic harmonic basis, led it many different directions. Wagner and Strauss pushed to the extreme the tension of prolonging chromatic pitches, whereas other composers, such as Debussy, overstep the boundary and moves towards modality. Composers of the Second Viennese School - Schoenberg, Berg, and Webern - completely eradicate any diatonic basis with atonal and dodecaphonic (twelve-tone serialism) harmony, and can thus be said to have moved through and beyond chromaticism.