Short answers, pulled from the story.
Relative keys are a major scale and a minor scale that share the same key signature, meaning they use exactly the same set of notes but begin on different tonic notes. For example, C major and A minor both have no sharps or flats, and F major and D minor both have one flat (B flat).
The relative minor is built on the sixth degree of the major scale. Count up six steps from the major tonic to find the relative minor's tonic. Equivalently, the relative minor starts three semitones below the major key.
The relative major is built on the third degree of the minor scale. The relative major's tonic sits three semitones above the minor key's tonic.
The most reliable clues are the first and last notes or chords of a melody, which usually land on the tonic. A raised seventh degree is also a strong indicator of the harmonic minor scale. For instance, frequent use of G sharp in music with no key signature points to A harmonic minor rather than C major.
Relative keys share the same key signature but have different tonics, such as C major and A minor. Parallel keys share the same tonic but use different key signatures and different notes, such as C major and C minor.
In German, the relative key is called Paralleltonart and the parallel key is called Varianttonart, which is the reverse of English usage. Most Germanic and Slavic languages follow this German convention, while Romance languages use different systems again.