The Use of Roman Numerals in Music Theory
Roman numerals represent the steps of a music scale, either major or minor.
Roman Numerals in a C major Scale
From each step of the scale, a triad or a basic chord in root position can be built. These triads are also called "Diatonic" (through the key).
The chord on the first step will then be called I, on the fourth IV, and the fifth V, and so on. The I, IV and V chords in any major or minor scale are the most important. They are the main chords of the scale.
Why not use regular Arabic (1,2,3 etc) numbers?
there are a lot of other numbers involved when playing chords, for
example the "number label" for seventh chords (7); to avoid confusion, roman numerals are instead used to represent steps on the scale, as well
as the chord built on that step.
This also makes it easy to
“translate” a piano chord progression to any scale you want. You
just need to know the scale you want to use, and then find each chord
from the steps in that particular scale.
And why not only use “regular” chord markings, like Gm7? (=G minor with a seventh)
Well, those tell you to play in a specific tonality- while the roman numerals chords make it easy to play chords from any scale.