The term Diminished is a loose term I always thought. Some people think of it as nothing more than a b5. There is a diminished traid (1 b3 b5), and the "diminished chord" (1 b3 b5 bb7), and the half-diminshed chord, also known as the m7b5 chord (1 b3 b5 b7).

The diminished chord (1 b3 b5 bb7) is a symetrical chord. IOW, it's created with a series of consucutive minor 3rd steps. Once you go three consecutive minor 3rd steps, you reach the root again. So, there's really no more notes you can build on top of it, in general diatonic chord building theory that is.

So, if we take a Edim chord, remember the term diminished chord means four notes, not three or anything higher than four notes, we get E G Bb Db. But since it's symentrical, either of these notes could be the root note, like this:

Edim = E G Bb Db
Gdim = G Bb Db E
Bbdim = Bb Db E G
Dbdim = Db E G Bb

So, the term "root" in this chord is fairly obsolete, except for communicating what note/root to build a particular chord from. IOW, the roots of each of these possible chords can be thought of as "enharmonic", or the same thing basically.

So, generally speaking, if a song calls for a Edim, you could more than likely play a Gdim, or a Bbdim, or a Dbdim. Because each of those chords has the exact same notes in it.

Some places where dim chords are used:

In a I ii iii ii I progression you can use a dim chord between each chord, as in the key of C here:

C C#dim Dm D#dim Em D#dim Dm C#dim C

This really spices up a iii7 ii7 Imaj7 progression, like this in the key of G:

Bm7 Abdim Am7 Gbdim Gmaj7

This is something similar to how Django and other Gypsy Jazz players would do this progression, and Jazz in general really.

Remember that even though I just say "dim" I am meaning a four note chord, not a triad. A traid might be written as a mb5 chord, like Amb5. This means 1 b3 b5. Hope that makes sense.

Now since the dim chord (4 notes) is symetrical, and the root is flexible and you'll find that there are really only three possible dim chord, and the rest are enharmonic chords of those three, persay.

Like if you play diminished chords chromatically you'll start reaching the enharmonic chord of the first at the fourth chormatic chord. Hmmm, hope that made sense. IOW, if I do them chormatically:

Edim Fdim F#dim Gdim G#dim, etc...

Once you hit the third chord Gdim you are playing the enharmonic chord of the Edim, and to continue...the G#dim is the enharmonic chord of the Fdim, and so on...

Hope that helps.