Triads are 3 tone chords built by superimposing intervals of 3rds. There are 2 types of 3rds: Major and minor. The Major 3rd is an interval equal to 4 half steps. The minor 3rd is an interval equal to 3 half steps.
With these 2 types of 3rds, we can have 4 combinations: 4-3, 3-4, 3-3, and 4-4. These are the Major, minor, diminished, and augmented, respectively.
As with all parallel naming, the Major is our point of reference [for triads = R-3-5]. Since in the minor, the 3rd is a ½ lower when compared to the Major 3rd, we call this a flat 3rd.
Similarly, the 5th in the diminished is a ½ lower, therefore, it is a flat 5. The diminished triad, when compared to the minor, has a lowered 5 while the 3 is the same [flat 3], so we can call this minor flat 5 [this is sub-paralleling - comparing to the minor, or something else that has already been modified]. The diminished triad is indicated with a o sign. Example, in C Major, the viio chord is Bo.
The augmented has a sharp 5, since its 5 has been increased from what is 'normal' [Major]. By increasing the 5, we have 8 half steps between the root and the 5th, rather than 7. The augmented triad first appears as a derivative in the Harmonic minor [chord space III]. The augmented triad is indicated by using a + sign. Example, in A Harmonic minor the III+ chord is C+.
We can build chords using other intervallic schemes, such as 4ths [quartal], but we don't typically call them triads. We reserve the term triad for 3 tone chords built by superimposing 3rds.