Functional Harmony: Tonic, Predominant, Dominant
Grouping all seven diatonic chords into three roles that explain why progressions feel like they go somewhere.
Every diatonic triad belongs to one of three functional groups, regardless of its Roman numeral: tonic (rest), predominant (setup), or dominant (tension seeking resolution). Progressions built from this T-PD-D logic are what make harmony feel like it's going somewhere instead of wandering.
Tonic function: I (and its stand-in, vi, which shares two notes with I and can substitute for it). Dominant function: V and vii° (vii° is essentially a V7 without its root, and shares the same tritone, so it pulls the same way). Predominant function: IV and ii (both contain the subdominant note and lead naturally into the dominant).
This gives you the basic tonal sentence: Tonic -> Predominant -> Dominant -> Tonic, or T-PD-D-T. I-IV-V-I and I-ii-V-I are both instances of exactly the same underlying shape, just with a different predominant chord. Once you hear the shape, you'll notice it everywhere — it's the skeleton under an enormous share of tonal music, classical and popular alike.
Russian classical harmony teaches the same territory slightly differently: instead of a distinct "predominant" category, ii is usually folded into a subdominant group alongside IV (both labeled S / С — субдоминанта), so the model reads as T-S-D-T. Same chords, same logic, one fewer named category.