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When would you say a piece, or a movement in a symphony, is in A minor versus C major? They both use the same notes on the scale... what would indicate that the piece is in A minor?
Let's take an example. The 2nd movement of Beethoven's 7th symphony. The principal key is A minor. But what distinguishes the key from C major in this movement? (Obviously, the mood is more appropriate to a minor key, but I'm sure there is a better answer.)
The last chord harmony of most pieces give a feeling of ending. (It would, wouldn't it - otherwise the piece goes on, potentially).With no key signature, shown, a piece could be in C major or A minor. This last chord gives a big clue as to which key the writer thinks it's in. The presence of G#, showing usually a V-I cadence is also a good clue, except that it is used in C major, to modulate to the relative minor. Maybe if it happens a lot, there is more chance of the tune actually being in A minor. Trouble is, that G# may show melodic or harmonic minor tendency, but wouldn't be there if the tune is written using A natural minor notes.
Another pointer is where the tune tends to gravitate to, especially at cadence points. If it feels like it's coming to rest on more Cs, or more As, there's the answer.
A nice example of this dilemma is 'Fly Me To The Moon'. Starts on Am, but ends on C.
Bottom line is probably - 'Does it really matter?'
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.
The other answers have good detail on the theory of the chords and scales, so I decided to give an example.
You mention the second movement of Beethoven's 7th Symphony as an example. There are no sharps or flats in the key signature, so how would you tell whether it's in A minor or C major? Let's take a look at that, shall we? Here's the opening chord:
Notice the notes that are being played (remember that the Horns transpose down a sixth, and the Clarinets transpose down a third). Horns: E; Bassoons: A,C; Clarinets: E,A; Oboes: C,E. This is a straight-up A minor chord being held for the first two measure of the piece. That's usually enough of a clue to let us strongly suspect that the piece is probably in A minor.
But we can go further. Lets look at the opening theme, which starts in the low strings (remember that the viola is written in the C-clef, so its first note is E). You can figure out the chords that each set of notes makes, but I've already written the chord name in blue under the staff each time it changes. I've also marked non-harmonic Passing tones and Neighbor tones. (In some cases this is a bit subjective, since many of them do create actual chords. For example, the first set of passing tones in the Viola creates a D7 in second inversion. For our purposes we don't care about this level of detail, though.)
Ignoring the first two measure of silence (while the woodwinds are playing that opening chord) this breaks into 4 phrases of 4 bars each.
Phase 1: Am-E-Am. This is a classic i-V-i. Other answers already explain why the G♯ is used to make E major, so I won't dwell on that. If we were in C major at this point, we'd be focusing on the vi chord and its secondary dominant.
Phrase 2: Am-C/G-G-C. Now this looks like a regular I64-V-I in C major. It seems as if Beethoven is trying to confuse us here, and make us think that we are in C major. These three chords can be said to tonicize C major (they temporarily treat C as the tonic). Does that mean we're in the key of C major? Not really, because keys tend to have a longer scope than three or four bars.
Phrase 3: C-B-Bm-A-Am. Suddenly, we are in what seems to be a tonally-ambiguous section, as the first cellos begin a descending chromatic scale. None of these chords (B, Bm, A) occur naturally in either C major or A minor. However, an A minor tonality is still arrived at by the end of the phrase, when the C♯ descends to a C♮. (A more detailed analysis might notice that, in the B minor measure, the G# that I labeled as a passing tone in the viola creates a G#dim chord (vii dim, relative to A), which has a dominant function leading back to A.)
Phrase 4: Am-E-Am-E-Am. This phrase is very similar to the first. It involves alternating Am and E chords (i and V) and gives us our first true cadence in the key of A minor, which brings a sense of finality and confirms that, yes, we are indeed in the key of A minor.
As an interesting (but largely irrelevant) side note, the fact that the F is always sharped in these phrases, even when the G is not sharped (as occurs in phrase 2), leads me to think that this piece actually leans more towards A-Dorian side of the minor key, rather than A-Aeolian (either way, it's still seen as A minor).
The tonic in C major is C major; the tonic in A minor is A minor. The key signature narrows it down to one of the two. An imperfect method to distinguish between a major key & its relative minor is to analyse the "chord" created when the main theme resolves. In the case of Beethoven's 7th, movement 2, this occurs, for example, just before the violins come in (26th measure). The "chord" created by the cellos & violas is an A minor.
First of all, it probably doesn't matter much.
Then, there can be clues.
When a piece is C major, it's quite likely that the main chords will be C major (tunica), G major (dominant) and F major (subdominant).
When a piece is A minor, it's quite likely that the main chords will be A minor (tunica) and D minor (subdominant), sometimes also E minor (dominant).
The difference is, as you said, the mood, which is caused by those clues.
Examples:
One way to quickly tell if a piece is in A minor versus C major is to scan through the score and see if you see a number of occurrences of G#. That's an indication of A melodic minor.
The answer varies according to styles. You seem to specify classical movements of symphonies and the like, so I'll answer it there. Jazz has so many harmonic nuances, and Pop/Rock really blurs the lines with natural minor.
In Classical theory, minor key does use the same key signature as the relative major, so A minor and C major have no sharps or flats. However, most classical works do not rely heavily on natural minor, which is all white keys. Instead, composers use melodic minor (F and G are sharpened), and harmonic minor (only G is sharpened). For other keys you have to learn some transposition skills, and key signatures, buts that's a different question.
That's visually studying the score. In terms of listening, the idea of key in all western music is that the music sounds like, or feels like it is pulled to one of the 12 keys in the octave. So for A minor, the music is naturally pulled to A, in C major music is pulled to C. In C major, The destination is C, which is called the tonic, or I (one). Look at the last note in the movement or piece. Sing it. It is probably your key, because that note is the destination note, tonic note, or "key" note.
The chord progressions revolve around A minor triad instead of C major triad. In other words, the A minor is the chord that is one or more of the following:
Additionally the "G" note/tone does not play a big role of emphasis for the melody or improvisation, but "A" and "E" tones do. In other words, melody or improvisation tends to target the root and fifth of the tonic chord.
When would you say a piece, or a movement in a symphony, is in A minor versus C major? They both use the same notes on the scale... what would indicate that the piece is in A minor?
This is no entirely true a minor has a G sharp that C major has not. There are some other indicators. If a piece has a anacrusis (Upbeat) than this is usually built on the dominant chord. SO if the upbeat was build around a E major chord than it would clearly indicate the key of a minor.
In the example you mention we start on C major chord which has a natural G which indicates that piece starts on the tonic chord of the key of C Major. Always check whether the leading tone of the minor key is raised or not. That should give you a clear enough indication most of the time.
If an appreciable number of cadences come to rest on an A minor chord, and E7 (with its distinctive G# note features heavily in the piece), it's in A minor. Don't be fooled by a final excursion into C major, or even a final A major chord.
Don't worry too much about attaching labels. It's good that you realise C major and A minor are closely related.
If this is about answering "theory exam" questions, just look out for the G sharps!
Note that the question of whether a melody is in a major key or its relative minor isn't always answerable, because it's possible for a collection of notes to be both a major-key melody and minor-key melody, with the notes playing different roles in the two keys. Consider, for example, a melody that alternates between A and F a few times and settles on Bb, then alternates between G and E a few times before settling on A.
If the melody was in F major, it would be going between the third and tonic, which are also the third and root of a I (F) chord, and then sits on the fourth, which would be the root of a IV (Bb) chord. Then it would cycle between the second and seventh, i.e. the fifth and third of a V (C) chord before settling on the third of the key and the I chord.
If the melody was in D minor, it would be going between the fifth and third, which are also the fifth and third of the I (Dm) chord, then sit on the sixth, which is the third of a IV (Gm) chord. Then it would alternate between the fourth and second, which are the seventh and fifth of a V7 (A7) chord before settling back to the fifth (which could either be the root of a V7, or the fifth of a I chord).
In the absence of any harmonization, if someone was asked whether the melody was in D minor or F major, neither answer would be uniquely right nor wrong. Many melodies will have features that would "fit" one key but not the other, such as a minor-key leading tone which would generally represent a rather awkward raised fifth in the relative major. Sometimes, however, a melody may be genuinely ambiguous, with every note having a plausible role in either key.
