C major is the relative major of A natural minor, which starts 3 notes below, or 6 notes above (same thing). Both of these keys use exactly the same notes:
C major: ---C D E F G A B C
A minor. A B C D E F G A
We can use exactly the same key signature to have these notes. That is why these two keys are "relative" to each other. They share the same key signature because they use the same notes. More often you'll see harmonic minor, which has a G# because the semitone from G# to A pulls more strongly to the tonic. Since that G# is not in the key signature, you will see accidentals on the 7th degree note. Therefore, if you wonder if the music is major or minor, figure out the relative minor (count down 3) and see whether the 7th leading to that note is being raised throughout most of the piece, using an accidental.
Music in minor mode will often end in a V-I cadence in that minor key. So if it's in A minor you may see E7-Amin, but if it's in C major you'll see G7-C. Minor keys have a different quality, often with a more sad or blue sound to them. The tonic chord is minor, while the tonic chord of a major key is major.