Answer: Lots of questions here. There's a discussion of all this in Chapter VI of Exploring Theory with Practica Musica, with examples, but let me give you the short version:

Natural minor is the same note-pattern as the major scale, but starting in a different place. "A minor" in its natural form uses the same notes as C major, but starting on A. A major scale has the Wholestep-Halfstep pattern WWHWWWH, and the natural minor has the pattern WHWWHWW.

But if you use only the notes of the natural minor there's no way to make a major chord on the fifth note of the scale (the dominant note), and generally that chord needs to be major. So, for harmonic reasons you'll often see the seventh note of the minor scale raised a half step. In A minor, the G becomes G#, for use in the E major chord E, G#, B. That's the harmonic minor.

But raising the seventh note that way makes an uncomfortable interval between the 6th and 7th notes. In A minor that would be F to G#: it's an "augmented second." That generally was thought exotic and hard to sing, so composers would raise the 6th degree too, turning A minor into A, B, C, D, E, F#, G#, A. That's the melodic minor: the upper body of a major scale grafted onto the lower half of a minor scale.

In reality all three forms of the minor may be found in the same music. To use again the example of A minor, the G (the 7th degree of the scale) tends to be sharped mostly if it's needed for harmony, and when it's not sharped then the F# doesn't get sharped either. An ascending melodic line is more likely than a descending one to have both those 6th and 7th degrees sharped, and that's why theory books (even mine) say there are two forms of the melodic minor: ascending and descending, with the descending one being just the natural minor again. This is a useful fiction, I suppose. But it mostly means that the minor scale is variable.

Before the needs of harmony reduced us to two basic modes: major and minor, there were others. The easiest way to know these is to consider the white keys of the piano. Just as we started on A to make "natural minor," also known as "Aeolian mode," so also if we start on D we get Dorian, E produces Phrygian, F makes Lydian, and G Mixolydian. "Locrian," starting on B, is a fictional scale invented by theorists to fill the gap between A and C, but in the modern era is often treated as if it had a natural birth. Locrian has no perfect fifth above its tonic, which is why it was never really used as a scale traditionally. But all the modes gradually evolved into major and the various forms of minor once the logic of harmony took hold (that's another story). In folk music you can still hear the remnants of modal music in, for example, fiddle tunes that have a major feel but with a lowered seventh, as in Mixolydian.

In music "parallel" doesn't have quite the same meaning as it does in geometry. The parallel major of A minor is A major. The parallel minor of C major is C minor. It just refers to the other scale that starts on the same note. But the "relative" minor of C major is the minor scale that uses the same notes as C major, that is, A minor. And going the other way, C major is the relative major of A minor. The relative minor of a major scale always starts on the sixth note of the major scale. The relative major of the minor scale starts on the third note of the minor scale. Relatives live near each other; parallels in the same place.