Tetrachords fall into the category of interesting thing to know, but really serve little practical purpose in playing music…
In a nutshell, a tetrachord is a series of four notes that, combined, span a perfect fourth (five half tones). The concept originated with the Greeks (i.e. the name that is Greek for four strings – as in a lyre) – who defined musical intervals as ratios (Pythagoras was a pioneer of this type of musical thinking…. And you thought he only did triangles.)
Any four note combination within that five half ton span is considered a tetrachord, but my guess is that your instructor has you dealing with major tetrachords (whole step, whole step, half step – or WWH). If that is the case, then if we start on the root of C, the tetrachord would be C, D (whole step), E (whole step) and F (half step)..... (CDEF).
If we then skip a whole step (to G) then the next tetrachord would be (GABC). Skip another whole step, and then you get (DEF#G)… again (ABC#D)…… so, our series of tetrachords looks like this:
(CDEF) (GABC) (DEF#G) (ABC#D) … and so on.
Notice that the first two tetrachords combined give you the C Major scale. Tetrachords 2 and 3 combined give you the G Major scale…. 3 and 4 give you the D Major scale……. And so on, all around the circle of fifths until you get back to CDEF.
I hope that this helps. If you keep going, you should end up with 12 tetrachords that include all of the sharps and flats.
) I use a simple music writing programme called Melody Assistant - If I want to hear how a tune goes, I enter it into MA and then press play. The problem comes if I try and read music myself without MA. I basically end up playing the right notes, but never at the right durations. I am hopeless at counting beats. I failed Math at school (but I speak 5 languages fairly fluently) - go figure. 
