The add2 chord is very similar to the add9 chord; the notes are in fact the same, but the difference is that the add2 and the add9 note belong to different octaves. You probably know that a common major chord includes the first, the third and the fifth note in its scale. By adding the second note, you get an add2 chord.
So if C major is used as an example, what happen is what C, E and G changes to C, D, E and G.
Notice that sometimes are the exactly same fingerings used as for add9 chords. The ambition is to present two different chords with the correct order of notes, but you will be okay if you use an Add2 chord instead of Add9 or vice versa.
To explain the difference between we could compare Eadd2 with Eadd9. The added note is in this case F#. In Eadd2 the F# is played as the third tone when strumming down strokes. In Eadd9 the F# tone is played as the last tone when strumming down strokes. The difference is in other words small, you can hear some change in the timbre, but if you feel you don't want to memorize both shapes it's alright to use the same shape in all situations when any Eadd chord is indicated.
Sometimes these chords are just referred to with the number two in the chord name, as C2, D2, E2 and so on.
Minor add chords
Some less common is the minor add chords.
The minor add2 is close related to the minor add9 and are in a sense equivalent. Yet, if you are meticulous with the sound, you probably want the added tone closer to the root note when choosing minor add2 instead for minor add9.
Chord progressions with add2 chord
Some examples of progressions using add2 chords:
C - Cadd2 – Fadd2
Aadd2 – D6 – Emaj7
Em – Emadd2 – Am – Amsus4
Existing, but quite uncommon is the related add4.
Some examples in short notation: