“Add” is used in chord symbols in certain cases where the usual assumptions don’t apply.
Strictly speaking, a chord with an extension (9, 11, or 13) contains the triad indicated by the note letter (root, third, and fifth), the seventh, the interval of the extension, and all lower extensions. (See here for the complete picture.) A ninth chord has no lower extensions, so a Cm9 is
C E♭ G B♭ D. The seventh (B♭) and the ninth (D) are both important to the quality of this chord. But a composer might not want the B♭ to be played. How is this solved? The symbol becomes Cm(add9).
The same applies to higher extensions. Strictly speaking, a Cm13 chord is
C E♭ G B♭ D F A, though it’s rarely played that way in practice. Musicians may choose to eliminate the fifth (G), the ninth (D), and/or the eleventh (F). That leaves the most important qualities of the chord in place; the root, the minor third, the seventh, and the thirteenth. Again, if a composer wanted an A in the chord without a B♭, the chord could be written as C6 or C(add13). By convention, a six chord is a triad plus a major sixth interval, so “add” would be redundant, but “add” is necessary in C(add13) to indicate that no B♭ is played. (The “add13” suggests a voicing where the A is an octave higher than 6, thus C6 is a far more common chord symbol.)
“Add” is also used for minor and major seconds, and perfect and augmented fourths when these aren’t meant to be suspensions. For example, Cadd4 is
C E F G whereas
C F G is Csus4. Confusingly, some publishers and composers use C4 to mean Cadd4 while other use it for Csus4. Obviously, it’s better to be explicit when you can.
When chord symbols get too complicated, they lose their utility. They’re meant to make things easier, not more complicated. In cases where specific voicings are desired, it’s better to write in standard notation.