Relative and parallel keys, and the two ways major and minor are related
A minor key can be related to a major one by its key signature or by its tonic; they are different relationships, with different uses, and confusing them is the commonest mistake in reading a signature.
Every key signature serves two keys, one major and one minor, and every major key has a minor key starting on the same note. They are two different relationships, and both get called "the minor of C" in conversation. Telling them apart is what lets you read a signature without error and change colour in a song without changing centre.
Relatives: the same signature
C major has no accidentals. A minor has none either. The two scales use exactly the same seven notes; what changes is which of them is the tonic.
A minor is the relative minor of C major, and C major is the relative major of A minor. The rule for finding the relative minor is to go down a minor third from the major tonic: C to A, G to E, F to D. For the relative major, up the same minor third.
That is why a signature alone does not give the key. One sharp can be G major or E minor; three flats can be E flat major or C minor. Whoever reads the signature has two candidates, and decides by the rest: the note the music ends on, the chord it starts with, the presence of a raised leading tone.
E minorParallels: the same tonic
C major and C minor start on the same note and are different keys: C minor has three flats, C major none. What they share is the centre; what they swap is the material.
C minor is the parallel minor of C major, and the name comes from their sharing a note name. The two scales differ in three notes — the third, sixth and seventh degrees, all a semitone lower in the minor — and it is that three-note difference that makes the major sound open and the minor closed over the same tonic.
What each relationship does
The two relationships serve different purposes, and the ear feels them in different ways.
The relative shares the material and changes the centre. A song in C major that moves to A minor uses the same notes; what changes is where they gravitate. The passage is smooth, almost imperceptible, because no new note appears; many songs go from a major chorus to a second section in the relative minor without a border being noticed.
The parallel keeps the centre and swaps the material. A song in C major turning to C minor for a stretch stays in the same place and changes colour abruptly, because three notes drop at once. It is a cheap and very effective device: the same melody, the same bass, and the whole mood darkens.
The same phrase, first in C major and then in C minor. Nothing moved; the material changed.
How to tell which is which
Faced with a key signature, ask two things. Which major key is it, and which minor starts a minor third below: those are the relatives, the two candidates. Faced with a song that changes mood without changing bass, ask whether the material changed over the same tonic: that is the parallel. The relative is a question of reading; the parallel is a question of colour.
What to take from here
Relatives share the signature and differ in centre; parallels share the centre and differ in material. The relative minor sits a minor third below the major tonic, and it is what makes a signature ambiguous. The parallel minor lowers the third, sixth and seventh degrees, and it is what lets you darken a passage without leaving the spot. Minor key harmony and modal interchange are what gets done with those two relationships.