Transposing Instruments
Transposing instruments are so called because they are notated differently from how they actually sound. On a piano or violin, what you see is what you get; i.e. when it says C, you play C, and C is what comes out of the instrument. Non-transposing instruments are said to be "in C".
However, certain other instruments are "in B flat" or "in E flat", which means that those are the notes that sound when playing a notated C. (Whether it's a lower or higher B or E flat depends on the instrument.)
Instruments are usually written as transposing instruments when they're part of a large family of instruments with similar construction but different registers, e.g. the flute or the clarinet. The intention is that the same written note should correspond to the same key combination on the instrument, so that a flautist can double on piccolo and alto without having to relearn any fingerings. The composer does the thinking so that the player can simply concentrate on playing.
In other cases, the instrument is written differently for typographic reasons, i.e. to minimize clef changes and/or ledger lines.
The Guitar in Staff Notation
In my guitar-centric world of music theory, transposing instruments would be superfluous information indeed—were it not for the fact that the guitar actually is one.
Guitar music is actually written one octave higher than it sounds, and for the latter reason of those stated above. The octave shift makes it possible to fit the range of the classical guitar onto one staff without the excessive use of ledger lines. Here are the open strings of the guitar when notated as usual:
Now compare that to if we were to write guitar music as it actually sounds, which requires the use of the bass clef, because of the comparatively low register of the instrument:
As you can see, the open top E string is already on the second ledger line. Most classical instruments have necks that join the body at the 12th fret, an octave above that, and electric guitars can have 24 or more frets. This would essentially mean separate clefs for rhythm and solos, lots of 8va/8vb or a second staff of ledger lines.
Tuning Down
Nowadays, many bands tune down their guitars for a darker and heavier sound, and/or to put the music within the range of their vocalist. This varies greatly, but can be as subtle as a half-step and as dramatic as a perfect fifth (the A below the low E).
When all six (or seven or eight) strings are retuned by the same amount, the guitar becomes a transposing instrument more akin to the first category mentioned above. It is still written as if in standard tuning, but the actual sound will differ from the notation. The amount of downtuning is indicated above the staff, either by stating the pitch of each individual string, or by simply writing "tune down 1½ step". This is to ensure that the notation corresponds to the correct fingerings. Thus, a guitar that is downtuned by a minor third sounds a minor tenth lower than notated.
"Dropped" tunings, where only one string is downtuned (usually the low E string by a whole step), are notated as per the majority of the strings; the range of the guitar is extended. Likewise with seven- or eight-string guitars. 12-string guitars are notated as six-string guitars, as the strings are played in pairs.
Terminology
It is a common and in its way logical misconception that the guitar is "in E", with progressive downtuning being "in E flat", "in D", "in D flat", and so forth.
This is logical in the sense that the outer pair of strings are both E strings, but it is plain wrong according to the principles of music theory, because a guitar in standard tuning is in C. Written C corresponds to sounding C, even though they're an octave apart. If a guitar were really in E, it would sound a major third higher or minor sixth lower than notated.
However, since the underlying principle of music theory is ease of communication, I would recommend to the reader to continue the previous practice in the interest of clarity.
