An edition of music based on critical examination of primary sources, like a composer's manuscript(s), a copy by someone close to the composer, or a first published edition. Music can be like the telephone game: what's published "down the line," sometimes years after the composer wrote the work, may be very different from what was originally written. A critical edition tries to represent the composer's original intentions as much as possible.
When searching for critical editions, keep in mind that they may be called different things:
How do you know what the critical editions are for Haydn and Mozart, or any composer? They're listed in Oxford Music Online, right before composer's work list at the end of the entry. There's also a list of critical editions, alphabetical by composer, included in the entry for Historical Editions. These are the best places to look, since they tend to be comprehensive; you can also find critical editions listed in IMSLP and Wikipedia, although sometimes these are incomplete.
A composer's music can be counted or numbered in a few different ways - all at the same time. You might have noticed this for Haydn: his pieces often have three different numbering systems. For example, the Haydn string quartet included in this research has:
And that's not to mention all the nick-names and subtitles that can be applied to classical music! For example, here's the Haydn string quartet in Wikipedia:
There's also a nice list of classical music nick-names, subtitles and non-numeric titles on Wikipedia.
My recommendation? Look up the piece you're researching in Oxford Music Online so you can see all the different ways it's numbered.
You can often find books that discuss, analyze and provide historical background and context for a composer's works. Most of the time you won't find a whole book written about one piece (although occasionally you will), but a book about a type of piece, like a composer's piano works, or symphonies, or string quartets.
Journal articles tend to be more focused than books, and you can often find an article about just one or two pieces.