A chord progression is the sequence of chords that accompanies a melody. There seem to be countless of them, but most songs use the same few chords, taken from one key. Understanding that logic lets you guess the chords of a song, transpose easily and write your own.
The seven chords of a key
A major key, for example C major, starts from the C major scale: C, D, E, F, G, A, B. Building a chord on each of these notes, using only notes of the scale, gives seven chords: C, D minor, E minor, F, G, A minor and B diminished (C, Dm, Em, F, G, Am, Bdim).
The pattern is the same in every major key: major, minor, minor, major, major, minor, diminished. In G major that gives G, Am, Bm, C, D, Em, F#dim.
Degrees in Roman numerals
To talk about a progression without tying it to a key, musicians number the chords with Roman numerals: I for the first, V for the fifth, and so on. Upper case means a major chord (I, IV, V), lower case a minor chord (ii, iii, vi), and the small circle a diminished chord (vii°).
So "I – V – vi – IV" means: the first chord, the fifth, the sixth (minor), then the fourth. In C it is C – G – Am – F; in G, G – D – Em – C; in D, D – A – Bm – G; in A, A – E – F#m – D. One progression, four keys.
Tension and resolution
The I is "home": the progression rests there. The V creates an expectation that calls for the return to I; it is even stronger with an added seventh (G7 in C). The IV moves away from home without as much tension. The vi, the relative minor chord, brings a more melancholic color while sharing two notes with the I.
This play of leaving and coming back is why some progressions feel "natural": the ear expects the resolution, and the music either gives it or delays it to surprise.
The progressions you hear everywhere
I – V – vi – IV: the pop progression above all, used in a huge number of songs since the 2000s.
I – vi – IV – V: the "fifties" progression, the one of slow dances and doo-wop.
vi – IV – I – V: the same chords as the pop progression, but starting on the minor, for a sadder tone.
I – IV – V: three chords, the basis of rock and roll, folk and blues. The 12-bar blues stretches it over twelve bars, often with seventh chords: I7 – I7 – I7 – I7 – IV7 – IV7 – I7 – I7 – V7 – IV7 – I7 – V7.
ii – V – I: the fundamental move of jazz, usually with seventh chords (Dm7 – G7 – Cmaj7 in C).
Finding the key of a song
Look at the chords of the song and find the key that contains them all. The first and last chords are often the I. A song in G, D, Em and C is almost certainly in G major (or in E minor, its relative, if it starts and ends on Em).
The app's Song tab does this for you: paste the chord sheet and it finds the key, writes the degree under each chord and suggests a capo position.
What about minor keys?
In A minor the seven chords are Am, Bdim, C, Dm, Em, F and G: the same as C major in another order, because A minor is the relative of C major. You often see i – iv – v or, with a major V borrowed from the harmonic minor scale (E or E7 in A minor), i – iv – V, which pulls back to the tonic more strongly.