This section is the most difficult to complete, so I strongly suggest getting in lots of practice with the intervals and chords lessons before delving into chord progressions. In this section, you must combine knowledge of intervals and chords with the ability to hear function within a chord progression.
Chord functions are labeled according to the scale degree of which the root of the chord is based. Each scale degree has a unique name. The scale degrees for a major scale are, in ascending order: Tonic, supertonic, mediant, subdominant, dominant, submediant, and leading tone (if the seventh degree is a half-step lower in minor keys, it is called the subtonic). So if you write out a C major scale, C would be the tonic, D the supertonic, E the mediant, and so on. These degrees are also notated using roman numerals, with upper case representing major and augmented chords and lower case representing minor and diminished chords. If you analyze the diatonic triads formed within a key (write out a major scale, then stack thirds on each scale degree to create triads), the roman numerals will be: I, ii, iii, IV, V, vi, viio (the leading tone chord is diminished).
Chord functions can also be grouped according to how they resolve or lead towards other chords. The tonic is the primary degree and normally represents the starting chord as well as the resolving final chord. Dominant chords serve the function of leading back to the tonic. The dominant (V) and leading tone (viio) scale degrees are the primary dominant function chords, although the mediant (iii) can also have a dominant function (albeit with a slightly weaker resolution to the tonic). Predominant chords function by generally leading to a dominant chord. The supertonic and subdominant degrees are examples of predominant functions.