Diminished chords are often the source of some confusion because they aren't defined to one particular type. They can be diminished, diminished seventh and half diminished. The main feature of these chords is that they are basically minor chords with a flattened fifth.
The first of them is the ordinary diminished. This is basically a diminished triad. It's the same as an ordinary minor chord but with a flat fifth.
Next is the half diminished. This is now more commonly called a minor seven flat five (min7b5) because as the name suggests, that's exactly what it is. It's the same chord type as the ordinary minor seventh but with the fifth scale degree flattened.
The last one is the diminished seventh, also sometimes called a full diminished. The construction of this chord is the same as the half diminished but with a double flat seventh! Yes you read that right, it's a double flat I.e. bb7. If you are thinking "but that's the same as a sixth" then you'd be right, it certainly is. So why not just call it a six. Well to be honest it's not that important. The theory behind it is too involved for a quick and easy explanation but the main thing about these weird kind of labels is it has more to do with written music and key signatures.
When it comes to writing music notation, this way makes a lot more sense. If you are not into reading and writing music, nor have much interest in music theory then it doesn't really matter whether you think of it a sixth or a double flat seven. It does however make sense to stick to the norm when you are writing or talking about it because it avoids confusion and keeps the pedants happy!
One thing to note about the diminished seventh is that it's intervals are all of an equal distance, a minor third (three semitones) apart. This makes the chord sound unstable, any of the notes in a diminished chord could be considered the root. For this reason, in a piece of music it would be the surrounding chords that would determine the key, or what note in the chord should be considered the root note.