Also known as "collected editions" or "collected works," complete editions are compiled by scholarly editors to most authentically represent the composer's intentions. The term "complete" may be inaccurate in some instances, as some editions are not yet finished, or some works which are historically missing are not included. These editions are found in the M3 call numbers.
You may find a piece in a composer's complete editions by locating their biographical entry in The New Grove Dictionary of Music and Musicians via Oxford Music Online, or its print version (available at the Music Library Desk). In the Article Contents navigation under the heading "works," you will find the titles of the editions for the composer, as well as important abbreviations that are used to differentiate between editions. In the examples below, JS Bach's editions and catalogs are listed.
When you search for a piece, you will see that the table in Grove tells you where you can find each piece in a specific edition. This example below demonstrates that you would find the JS Bach Motet "Singet dem Herrn ein neues Lied" in Band (volume) 39, page 5 of the Bach Gesellschaft (or BG). In the Neue Bach-Ausgabe (or NBA), the same motet is found in Series III, Band (volume) 3, page 3.
The numbers on the left side of the table under BWV and BC are from Bach's thematic catalogues - Bach Werke Verzeichnis and Bach Compendium. Thematic catalogues are indexes which organize a composers' works. This is used with many composers, such as Köchel (K) for Mozart or Deutsch (D) for Schubert.