Answer: The big difference between C major and A minor is that the "tonic" - the home note, so to speak, is C when in C major and A when in A minor. And that implies different harmony, too. Suppose that you see a melody that has no sharps or flats anywhere: no key signature, no "accidentals" placed in the music ("accidentals" are sharps or flats that are not in the key signature). But you can see that the melody ends on A. It's undoubtedly in A minor. Look more closely and you'll see that the chords outlined or otherwise implied will tend to follow progressions of chords where the chord roots relate to A. The most common chords in most music are those built on the tonic (the "I" chord), those built on V (fifth note of the scale), IV, and II. So, in A you'd see a lot of chords built on A and E and D. But in C major the important roots would be C, G, F, and D, especially C and G.
If accidentals do appear in A minor, the likely ones will be G# and possibly F#. The G# will be there because the composer will usually want to use a major "V" chord - the chord built on E, the 5th note of the scale - and that requires the G be sharped. If the melody rises by scale note to that G# the F might get sharped too, because that makes a major second instead of the more exotic augmented second F-G#. But notes raised for this purpose are not included in the key signature and that's why they need to be written in as accidentals - they're optional.
So, to generalize:
(1) First check the key signature, which will narrow the choice to two keys: a major key and its relative minor: C major and A minor, or D major and B minor, etc. The relative minor is the key three scale steps downward from the tonic of the major key.
(2) Then check the final notes and identify the root of the final chord, or the final note of the melody. It will typically be the tonic: C if in C major, A if in A minor, D if in D major, etc.
(3) When you are more used to looking things over at a glance, notice the harmony. If the piece is polyphonic the chords will be apparent. If it's just a melody they will often be outlined or otherwise implied (there's a section in Exploring Theory with Practica Musica that discusses how to identify implied harmonies). The harmony is really the clincher: tonal music will frequently use the V-I or V-i cadence, which tells you which note is "I," the tonic.
As for the second part of the question: I think you're asking how can you tell the number of sharps or flats expected for a particular key -? Start with C, which has no sharps or flats. For every fifth you move upward, a sharp is added to the key signature for that tonic: C, G, D, A, E, B, F#, C# have 0, 1, 2, 3, 4, 5, 6 sharps in their key signatures. For every fifth you move downward you add instead one flat to the signature: C, F, Bb, Eb, Ab, Db, Gb, Cb have 0, 1, 2, 3, 4, 5, 6 flats. And each major key has a relative minor that uses the same signature but whose tonic is three scale steps downward: for example, A minor, E minor, B minor, F# minor have 0, 1, 2, and then 3 sharps. You don't really need to refer to a "circle of fifths" or a fact book; just count the fifths up or down from C for major scales; count fifths up or down from A for minor scales.
