When they harmonize a melody, composers try to organize the chords in a pleasing way.
Let’s talk about some of the governing principles they use.

In this post, we will look at the the most fundamental mechanism: the cyclic progression. In a subsequent post we’ll look at sequences.
We will stick with diatonic harmonies, i.e. we will not discuss secondary dominants, or chromatic passing chords, or chromatic mediant and submediant chords, or borrowed chords. You can always add those, of course.

1. Major keys
Remember that for a major key the chord family is: I ii iii IV V vi vii°
Example: C Dm Em F G Am B°

The strongest and most conclusive harmonic movement is when the root of the current chord goes down a diatonic fifth, for example ii -> V (e.g. Dm -> G). This is called a cyclic movement.
When the landing chord is the I chord, in other words when we have V -> I, we have a “perfect cadence”.

Another, almost equally strong way to go to the tonic chord is by means of an ascending semitone movement, i.e. vii° -> I

Let’s start from the tonic chord and work our way backwards, applying cyclic root movements.
We just saw that the strongest possible way to get to I is from V or from vii°.
So we have (V, vii°) -> I.

The strongest way to get to V is from ii because then the roots also move by a descending 5th. The best way to get to vii° is from IV.
So we have (ii, IV) -> (V, vii°) -> I.

The best way to get to ii is from vi and the best way to get to IV is from I but we already have I in our list. So we have:
vi -> (ii, IV) -> (V, vii°) -> I

The best way to get to vi is from iii, so we have:
iii -> vi -> (ii, IV) -> (Vn vii°) -> I

And finally the best way to get to iii is from vii°.

We therefore have the following scheme:

The chords V and vii° are called dominant chords and the chords ii and IV are called predominant chords.

Let’s build a chord progression in C Major, starting from I.
Since I is the tonic chord, it can go to any chord; let’s go to vi.
C -> Am.
From vi we can go to ii of to IV; let’s go to ii:
C -> Am -> Dm.
From ii we can go to V or to vii°; let’s pick V:
C -> Am -> Dm -> G
And V leads us back to I:
C -> Am -> Dm -> G -> C

Of course, this also works with richer diatonic chords, for example: Cmaj7 -> Am7 -> Dm7 -> G7 -> C.

2. Minor keys
For minor keys the basic principle is the same. However, there is a slight complication in that there are two different minor scales that are relevant from a harmonic perspective: the natural minor scale and the harmonic minor scale.
These scale are: 1 2 b3 4 5 b6 b7 and 1 2 b3 4 5 b6 7 respectively.

Remember that chords are numbered with respect to the parallel major key. So, for example, if we want to write down the chords for the A natural minor key, we will take the A major key as the reference.
So, for the natural minor key we have: i ii° bIII iv v bVI bVII
And for the harmonic minor key we have: i ii° bIII+ iv V bVI vii°

The diagram for minor keys is as follows:

As you can see, the chords result from a mixture of natural minor and harmonic minor.
Let’s build a chord progression in Am.
Let’s first go to bIII:
Am -> C
From bIII we go to bVI:
Am -> C -> F
From bVI we can go to ii° or to iv; let’s go to ii°:
Am -> C -> F -> B°/D (half diminished chords are usually played in first inversion).
From ii° we can go to V or to vii°; let’s go to V:
Am -> C -> F -> B°/D -> E
…and we’re back to I:
Am -> C -> F -> B°/D -> E -> Am

3. Conclusion
The guidelines discussed above are just that: guidelines. This is of course not the end of the story and there are many more possibilities. But the guidelines are powerful and will often help you organize your chord progressions.