Aha. Now I see... I assumed that you knew about V-I. It's at the heart of classical music theory. In any key, V goes to I (except when it doesn't).
Let's take a look at classical cadences:
Plagal cadence: IV to I (or iv to i): In C major, F to C ... in C minor, Fm to Cm. This is the "Amen" at the end of church hymns. Also found in the blues and much rock.
Authentic cadence: V (or V7) to I (or i). So the tonic chord can be major or minor, but the V is always major.
C major: G (or G7) to C
C minor: G (or G7) to Cm
This is the most important cadence. Most chord changes in classical progressions are based on V-I changes. With many consecutive V-Is, you get the circle of fifths.
Deceptive cadence: V (or V7) to vi (or VII):
C major: G (or G7) to Am
C minor: G (or G7) to Ab major
The most common variation on V-I. It only works because V goes to I 90% of the time, making this change really jump out. But overuse kills its value!
So, the bottom line: when your chord changes drop a fifth every time, its a long sequence of V-I.
Whatever chord you're playing is the V of the next chord.
A is V of D (D E F G A, just count the letters), so some kind of D chord follows some kind of A chord. And so on.
The first half ends on Am because that's the key of the progression. That's where it started. So you repeat the first four chords from the first half of the chorus. Then it's a IV-V-I in D as a subdominant function, then IV-V-I in the home key of Am... except the last chord is major, a large-scale Picardy third.
That should clear everything up.