Nonharmonic tones, or non-chord tones, are notes that belong to the melodic aspect of a piece of music, not the harmonic aspect. They are called non-chord tones because they do not fit into the harmony or chord occurring. Although non-chord tones do not fit into the sonority they are played over, they instead have other effect on the music separate from harmony. This can be to smooth out voice leading in the form of a passing tone, to add tension in the form of an appoggiatura, or to achieve many other musical goals.
In short, nonharmonic tones are notes that do not belong to a chord that create the melody of a piece.