The different note values in musical notation (whole note, half note, quarter note, etc.) inform the musician of the duration of the note to be played. Adding a dot after the note tells the musician to play the note for its full value plus an additional half value. A whole note normally lasts for four beats. A dotted whole note lasts for four beats plus an additional two beats.
A double-dotted note lasts for the value of the note plus a half value of the note plus an additional half of the added value. For example, a whole note lasts for four beats. A dotted whole note lasts for four beats (original value) plus two beats (half of the original value). A double-dotted whole note lasts for four beats (original value) plus two beats (half of the original value) plus one beat (half of the additional value).
Each dot adds half the value of the dot added before it. Dots on notes can potentially continue ad infinitum, although notes with more than three dots are very rare. Some examples using whole notes:
Triple-dotted whole note: one whole note (four beats) plus one half note (two beats) plus one quarter note (one beat) plus one eighth note (half beat), for a total duration of seven and a half beats.
Quadruple-dotted whole note: one whole note (four beats) plus one half note (two beats) plus one quarter note (one beat) plus one eighth note (half beat) plus one 16th note (quarter beat), for a total duration of seven and three quarters beats.
Double-dotted whole note: one whole note (four beats) + one half note (two beats) + one quarter note (one beat), for a total duration of seven beats.
Double-dotted half note: one half note (two beats) + one quarter note (one beat) + one eighth note (half beat), for a total duration of three and a half beats.
Double-dotted quarter note: one quarter note (one beat) + one eighth note (half beat) + one 16th note (quarter beat), for a total duration of one and three quarters beats.
Double-dotted eighth note: one eighth note (half beat) + one 16th note (quarter beat) + one 32nd note (eighth beat), for a total duration of seven eighths of a beat.
J. S. Bach's "Fugue in D-Major" from Book 1 of "The Well-Tempered Clavier" for piano contains many dotted and double-dotted notes in both the right- and left-hand parts. Frederic Chopin's "Prelude in G Major" for piano uses double-dotted notes, as well as triple-dotted notes, in the right-hand melody. The first note of a Joseph Haydn string quartet (Op. 74, No. 2) is a double-dotted eighth note played by the first and second violins. Double-dotted notes continue to appear throughout the piece, played by all four instruments.