Which chords belong in a key, and what do the Roman numerals mean?
Each key owns seven chords, one built on each scale note; Roman numerals (I, IV, V, vi...) name them so patterns work in every key.
Every key comes with a built-in family of seven chords, made by stacking notes from its scale. In C major the family is C, Dm, Em, F, G, Am and one odd one out (Bdim).
Songs mostly pick from that family, which is why some chords "just go together": they are siblings from the same key.
Build a chord on each degree of the major scale and a fixed quality pattern appears: major-minor-minor-major-major-minor-diminished. Roman numerals name the degrees, uppercase for major and lowercase for minor: I ii iii IV V vi vii°.
Numerals are the transposable language of harmony: I-V-vi-IV describes the same song in every key, which is why chord charts, YouTube lessons and this trainer speak in numerals.
To harmonize a melody, favor chords whose tones include the melody note, and lean on the three pillars: I (home), IV (away), V (tension that resolves home).
Minor keys complicate the pattern productively: natural minor gives i ii° III iv v VI VII, but the harmonic-minor raised 7th converts v to a functional V, which is why minor-key songs still use major dominant chords.
Function groups the family: tonic (I, vi, iii), subdominant (IV, ii) and dominant (V, vii°) chords are largely interchangeable within their group, which explains substitution: ii-V-I works where IV-V-I does, richer.
Borrowing extends the family: the bVII in rock (from Mixolydian or the parallel minor) and the minor iv in pop ballads are out-of-key guests with in-key jobs. Numeral literacy is what lets you notice, name and reuse them.