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 notes belong to different octaves. A common major chord includes the first, the third and the fifth notes 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 that C, E and G changes to C, D, E and G.
Notice that sometimes the exactly same fingerings are used 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 Aadd2 chord instead of Add9 and vice versa.
To explain the difference between these, 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 downstrokes. In Eadd9 the F# tone is played as the last tone when strumming downstrokes. 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 is 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, F2, G2, A2, B2 and so on.
The Badd2 chord has no easy fingering. An alternative to the pictured chord is X2464X. Dadd2 can also be played as XX4230 (Dadd2/F#). GAdd2 can also be played as 300003.
A quite common sequence comes from varying Cadd9 with a G major. You could when for example add a D major and you get something like this (used in the Green Day song "Time of Your life"):
G – Cadd2 – D
More progressions, including add2 chords:
Cmaj7 - Cadd2 – Fadd2
Cmaj7 - Gadd2 – Fmaj7
Cadd2 – Em7 – Dsus4
Aadd2 – D – Dsus4 – Gadd2
Eadd2 – F#m11 – G#m7#5 – E6sus4 (see tab)
Existing, but quite uncommon is the related add4. These could also be identical with sus4 chords.
Some examples in short notation: