The three minors
Minor comes in three forms. They all share one key signature — they differ only in how the 6th and 7th degrees behave near the top of the scale. Here's why there are three, using E minor (one sharp, F♯) throughout.
The problem natural minor has
Play E natural minor up and stop on the 7th degree, D:
That whole step means natural minor has no leading tone — no strong pull home. Composers fix it by raising the 7th.
Harmonic minor — raise the 7th
Raise D to D♯, a half step below the tonic. Now there's a real leading tone, and the chord built on the 5th degree becomes major and strong (that's the "harmonic" part — it's about the chords).
Melodic minor — raise the 6th too, going up
The augmented 2nd sounds fine in a chord, but jumpy in a running melody. So when the line is ascending, raise the 6th as well (C → C♯) to smooth it out. Coming down, there's nothing to smooth — no pull toward the tonic — so it reverts to plain natural minor.
Hear the difference
Try
On the A string: play C (low 2), then D (3rd), then E (open) — the top of natural minor, smooth and mild. Now play C (low 2), then D♯ (3rd finger stretched forward), then E — the top of harmonic minor. Feel the stretch, hear the leap, then hear how hard that D♯ pulls up to E. That leap is the price of the leading tone.
The usual mistake
Trying to memorise all three forms in all twelve keys — the source book warns against exactly this, and it's right. Learn the idea: one key signature, and the 6th/7th get raised or not depending on whether you need a leading tone (harmonic) or a smooth melodic line (melodic). Then drill the specific minor keys your pieces are actually in. For this stage, that's E minor.