They are made by four notes instead of three. We add a third to the triad chord, or in other words, the seventh of the chord. If the seventh added is Major (when it's half a tone away from the tonic if we invert it), the resulting chord will be with the seventh “Maj7”, and if it is minor (when it is at a distance of a tone from the tonic if we invert it), the resulting chord will be called the seventh (without indicating that this seventh is minor), and the seventh shall be written with no. 7.
Another block of tetrad chords can be formed from diminished triads: If the seventh we add to the diminished triad chord is minor, we have a chord that many call semi-diminished, but I will refer to this agreement in class as a: “minor seventh, fifth flat” or “minor seventh, fifth flat”.
If the seventh we add is diminished, the resulting chord will be called the "seventh diminsihed"
In the case of diminished chords it is allowed to harmonize some notes to make it easier to read them.
Lastly, I wanted to add a couple of chords that we often find in the course books in the modern syllabus. These are the chords of "sixth" and "sus 4". It's very simple: The sixth chords are major triad chords to which we add the sixth major. These are called sixth chords.
Finally, The chords “7sus 4” are seventh chords to which we substitute the third for the fourth.
Like the triad chords, the tetrad can also be inverted, adding a new position: the third inversion when the seventh is the lowest note in the chord.