Our first diminished chord, the minor 7th flat 5 chord, is also commonly known as 'half diminished'. This chord is a more relaxed version of the intense diminished triad and therefore finds a use in blues, jazz and a variety of other genres. We therefore tackle the construction of this chord as well as the uses.
Our first diminished chord is known as both "minor 7th flat 5" and "half diminished". We feel as though the minor 7th flat 5 is a better and more suitable description, as well as the most common way you will see it written out. If you follow the basic logic in the name "m7b5", we can see that it is essentially a Minor 7th chord, with a flattened 5th. A minor seventh chord os constructed 1st, b3rd, 5th, b7th. So all we need to do is then flatten the fifth to get 1st, b3rd, b5th, b7th. Voila!
To help you put this into practice, here is a common shape for the m7b5 in the key of E (as per the example above). Notice the notes on the neck as you go through it, trying to say the degrees of the scale (i.e. 1st, b3rd, b5th etc...).