To amplify on the diminished type chords:

The diminished triad is just the [1, b3, b5] chord, so called because the interval 1-b5 is a diminished interval. This is the one written as Cdim for example.

The chord you'll probably come across most often is the diminished 7th chord (dim7) [1 b3 b5 6(or bb5 as Sky-Coyote noted above)]. This chord can be created by taking a dominant 7th chord [1 3 5 b7] (like C7) and flattening every note except the root, thus dominant 7th=[1 3 5 b7], flatten every note except the root gives [1 b3 b5 bb7]=[1 b3 b5 6]=diminished 7th chord, often written as Cdim7 or Co7 or just Co and is the chord most often meant in standard chord sequences when Cdim is written. (The chord is useful because sharpening any one note of the chord produces a dominant 7th chord and thus depending on which note you sharpen, lets you go off in several directions).

The half-diminished chord is a minor 7th chord with a flattened 5th: [1 b3 b5 b7] and may be written as Cm7b5, Cm7-5 or again as Sky-Cotote noted as Cø7 or Cø.

Mick