The notes of Aadd9 are A, C#, E and B. A is the root, C# the major third, E the fifth and B the ninth. Said aloud, its notes are A, C sharp, E and B.
From the low string to the high one you hear A, E, B, C# and E. The note heard most is E, twice. It is not the root but the fifth. The lowest note is A on string 5 and the highest is E on string 1.
Aadd9 lets two strings ring open. String 5 gives A and string 1 gives E. String 6 is not played. Everything fits between frets 2 and 4. The second shape on this page starts at fret 2.
A is one change away. String 3 moves from fret 4 to fret 2, bringing in A, the root of A. It shares two notes with F#m, A and C#. Going to Bm, one position stays the same.
Aadd9 shows up in six keys. In A major it is the Iadd9, in D major the Vadd9 and in E major the IVadd9. The rest appear in the key list.