A is tonicised (the term we use) rather than C. That means that the tonic chord, the one that defines the key and resolves all harmonic motion, is confirmed by various means as A minor. Tonality is defined by a hierarchy of interval relationships, the strongest being the relationships of a fifth and a fourth. In the key of C, that would be G and F respectively, which is why you see chord progressions like C Maj - F Maj - G Maj - C Maj all the time. The equivalent progression in A minor would be A min - D min - E Maj - A min. I'll get to why E Maj in a bit.
Now those relationships aren't symmetrical: rising a fifth (or falling a fourth), for various reasons both physical and cultural, creates a greater strain than falling a fifth. (TL;DR is that in the harmonic interval of a fifth, the upper note and its harmonics reinforce the harmonics of the lower note, thus moving from one chord to the one a fifth above is changing the effective fundamental, whereas moving to the chord a fifth below supplies a fundamental that encompasses both chords.)
The net effect is that the fourth degree of the key (F Maj in the key of C Major, for example) is, in effect, a point of relaxation in a chord progression in that key (because moving from the tonic to the fourth degree is an easy downward motion of a fifth), whereas a movement to the fifth degree (G Maj in the key of C) sets up an opposition to the tonic, and falling from the fifth degree to the tonic resolves that opposition. That's why progressions like C - F - G - C are so common. In this case, C is being balanced by F and G, and the final motion G to C is a resolving one which confirms C as the tonic.
Melodic intervals aren't symmetrical in strength either. The strongest intervals melodically are seconds, and the very strongest motion is by a minor second upwards. In both C major and the natural scale of A minor, the minor seconds are found between E and F, and B and C. In C Major, when you move from a G Maj chord to a C Maj chord, you have not only got a strong root movement (establishing a fundamental of C), you've got the B in the chord of G (the third of the chord) pushing into the note of C in the tonic chord as well.
Given that the minor second intervals in both C major and natural A minor are the same, the chord on the fifth degree of A minor is normally going to be a minor chord, E min, with G as its third. The melodic motion from G to A is a major second, which isn't as strong as a minor second, so it's very common to substitute G♯ instead, which makes the chord on E a major chord. (If the melodic line was rising from E to A, it's not unusual to sharp F as well.)
So, if you look at a piece in C, you'll find that the piece usually begins with a C Maj chord or G Maj chord (not always, but these are most common), and usually ends with a G Maj chord followed by C Maj final chord. In A minor, the pieces usually start with an A min chord, sometimes an E Maj, occasionally an E min, and usually end with E Maj followed by A Maj or A min. (The reason for A Maj as the very last chord of the piece is that asymmetry I mentioned earlier: a major third in a chord is a better match to the lower harmonics of the chord's root note than a minor third, and thus the chord is more stable. The term for ending a piece in the minor mode with the tonic major chord is tierce de Picardie or Picardy third.)
You'll also see a fair number of G♯s (generally pushing into a neighbouring A) as SSteve mentioned, and possibly some F♯s as well. That is melodic motion inside the piece that is helping to confirm A as the tonic.