Roman numeral analysis, and the chord described by the place it holds
Roman numeral notation describes a chord by the position it holds in the key, not by the note it sits on; the case of the numeral carries the quality, and thinking this way is what lets you transpose a whole song in your head.
A chart says "C, A minor, D minor, G seventh". An analysis says "I, vi, ii, V7". The chart describes the chords by name; the analysis describes them by the place each one holds in the key. The second is the same in C, in F and in B flat, and that is why whoever thinks in degrees plays a song in any key without calculating anything.
The numeral is the degree
Each chord in the harmonic field takes the number of the degree it was built on, in Roman numerals: I for the first, II for the second, up to VII. In C major, the F chord is the IV because F is the fourth degree; in G major, the IV is C, for the same reason.
Both sequences read the same: I, ii, iii, IV, V, vi, vii°. It is the reading, not the names, that is learned once for all twelve keys.
The case carries the quality
The most useful convention in the notation is that the size of the letter says the chord's quality. Upper case for major: I, IV, V. Lower case for minor: ii, iii, vi. A small circle for diminished: vii°. A plus sign for augmented, rare in the major field: III+.
So the major field written in degrees already shows the qualities without stating them: I, ii, iii, IV, V, vi, vii°. Three upper case, three lower case and one diminished, in the positions the scale puts them.
Sevenths, extensions and bass
Added notes follow the numeral, as in a chart: V7 is the dominant with a seventh, ii7 the second degree with a seventh, Imaj7 the tonic with a major seventh. The half-diminished gets a slashed circle: viiø7. The bass, when it is not the root, is written with figured-bass numbers or, in popular practice, with a slash and the degree of the bass.
That reads ii7, V7, Imaj7, and it is the spelling of the II–V–I that all jazz reading uses. Note the 7 after the V: the dominant with a seventh and the dominant triad are different chords, and the analysis tells them apart.
Chords from outside
A chord that does not belong to the field gets a notation saying where it came from. A secondary dominant is written with a slash: V7/ii is the dominant of the second degree, V7/V the dominant of the dominant. A chord borrowed from the parallel minor is written with the flat that lowered its root — ♭VI, ♭VII — or with the case that changed: iv for the minor fourth degree.
I, V7/ii, ii7, V7, I; and I, iv, V7, I. The notation says not only which chord it is, but what role it plays: the A seventh is not "a chord from outside"; it is the dominant of the D minor that follows.
Why analyse
Three reasons. Transposing: a sequence in degrees is played in any key. Hearing: whoever knows a cadence is ii, V, I recognises it in any song, with any chord names. Comparing: two songs with completely different charts can have the same analysis, and then you find out they are the same sentence.
What to take from here
Roman numeral analysis describes a chord by its position in the key, in a Roman numeral, with the case saying the quality and the added numbers saying the sevenths and extensions. Chords from outside get a notation saying where they came from. It is the vocabulary in which harmony is transposed, heard and compared, and it is the language the other articles on this site were already speaking.