For example, if the key signature on the left doesn't have any sharps or flats then it could theoretically be in either the C Major scale or the A Minor scale. How do you determine which one?
If it starts on C and ends on C, it's probably in C. And vice versa.
If it has lots of A minor and E minor chords, it's probably in Am (E is the dominant). Likewise, lots of Cs and Gs implies C Major.
There are a few other indicators — for example, the harmonic minor scale raises the 7th one semitone (as Jenny mentions) and the melodic minor raises both the 6th and 7th while ascending — but largely speaking it doesn't matter. Frequently, they are two names for the same thing. Often people say that minor keys sound sad and major keys sound happy or triumphant, but this is nothing more that psychological tricks (see Why do minor keys sound “sad”?).
You can use C Major to write a sad song, and you can use A minor to write a happy song. In the end, the only way to know what key is being used is to ask the composer. The intent is the only thing that unequivocally distinguishes a major key from the minor with the same key signature.
If i want to determine the key of a song or part of a song. I'm looking for dominant chords and where those resolve to. If i see a G7 (and especially when it resolves to C chord) - then i suspect C major. For a minor often this will be E7 (in place o Em7) resolving to Am chord.
This E7 is dominant chord from harmonic minor scale and is in common use.
This is not a 100% sure method, but i would say 90% :) Whether You need to determine key for jazz changes or for the pop tune.
P.S. Looking at the last note in melody can be another tip
In C-Major, the "main" chord of a song is usually the C-E-G chord, and that's usually the beginning and ending chord of a song. The other two most common chords in simple tunes are C-F-A and B-D-F-G. If the song uses those chords, then it's probably in C major.
If it uses a minor third instead of a major third in the first two notes of the primary chord (3 half-steps vs. 4 half steps), then it's minor. Play these on the piano and you'll hear the difference: C-Eb-G is minor, and C-E-G is the major chord. You can transpose up to A to make the comparisons there, too.
You look at the score. Between the Clef and the Time Signature there will usually be some sort of indication of the key. If there is no sharps or flats in between the clef and the time signature then you are either in C Major or a minor. (The Major key with no sharps or flats)
Then you need to look at whether there are any notes raised in the piece. The leading tone of minor keys is always raised by a semitone. So if you see a G# you need to think a minor.
- Think about the sharps they give.
- Determine the Major key that consist of those sharps.
- Determine the relative minor of that Major key.
- Look for the leading tone of the minor key.
- See if it is raised or not.
And remember the note that could be raised could be a flat that is raised to the natural note. It could be a c that is raised with the use of a sharp or it could also be a note with a sharp raised to a double sharp. So do not just assume if you do not see a sharp that there is no notes raised.
You're kind of overcomplicating this. The best [and easiest] way to find out whether it is C major or A minor is by looking at which note starts and ends the song. Sometimes, this doesn't work and you might feel a little lost. You need to look at which note is within the song more frequently then the other. That should help. I don't know if this is helpful, but thought I'd put my two cents in. :)