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When looking at the scale of A minor, you can form the following common chord progression:

A Minor Chords: I(Am) - IV(Dm) - V(Em) - I(Am)
This sounds a bit "even". We can change a single tone in the minor scale to give this progression more harmonic tension. If we make the seventh tone higher (the G will become a G#), we get these chords:

A Harmonic Minor Chords: I(Am) - IV(Dm) - V(E7) - I(Am)
The E7 in the second progression provides more tension, it has a larger "urge" to resolve back to A minor, hence the harmonic strenght has been enlarged.
In short: if you raise the seventh tone of a minor scale, the harmonic tension increases. A minor scale with a raised seventh tone is called "Harmonic Minor"
A-harmonic minor consists of the following tones:

It sounds like this:

A Harmonic Minor