The ‘harmonic’ and ‘melodic’ minor scales.

Before we tackle these, a little background information would help understanding them and where they come from.

From the 17th to 19th centuries, music was very structured and bound by some pretty strict rules, guidelines and methods. One of these was the use of something called a 'cadence'. Think of a cadence as a punctuation mark.
There are four types of cadence:
'perfect'
'imperfect'
'interrupted'
and 'plagal'
I'll not go into how why and where these are used as it is a large topic in it's own right [and only useful to folks formally studying music or writing in Baroque, Classical or Romantic style]. For this topic, we are interested in the 'perfect' cadence [a chord progression of chord V to I]. Think of a perfect cadence as a 'full stop' at the end of a section of music [also at the end of the entire piece].
Not only does a chord change from V to I produce a perfect cadence, but V must be major [better still, a dominant 7 chord].
If you cast you mind back to the chords generated by a major scale you'd notice that chord V7 is a dominant 7. So in the key of G major a perfect cadence would be D7 to G.
So let's look at this with respect to a minor key.
The chords of the key of Em are Em, F#mb5, G, Am, Bm, C, D.
So we have a problem because chord V is minor [in this case Bm].
V to I in the key of Em is Bm to Em and it just doesn't sound very 'final'.
Not good for something that wants to behave like a 'full stop' - lol

The solution is to modify chord V from being minor to major [or from a minor 7 to a dominant 7] when executing a perfect cadence in a minor key.
So in the key of Em, the perfect cadence chord progression of Bm to Em becomes B to Em [or better still B7 to Em]. Yummie, this sounds nice and final like a good musical full-stop should.
Now we have a new problem, the scale don't fit with the modified V chord.

The scale of Em is: E, F#, G, A, B, C, D
The modified chord V7 is now B7 which is: B, D#, F#, A - so we can see that the D from the scale and the D# from the chord clash.
Clearly this will sound - in technical terms - totally pants!!!
Ouch!!

The solution is to modify the scale to fit the modified chord [but only during the moment of the cadence - not for the whole piece of music].

This of course gives birth to a new scale that contains E, F#, G, A, B, C, D#…
The spelling of this new scale is:
Tonic
major 2nd
minor 3rd
perfect 4th
perfect 5th
minor 6th
major 7th <- here is the modified note: The major 3rd of chord V7, also the major 7th of the modified minor scale.

This modified scale is called the 'harmonic minor' scale
it sounds very cool too

This is the scale that you'll often hear Malmsteen tearing up and down as fast as he can over chord V of a minor key.

So, back to the 1700's in the key of Em at a perfect cadence.
If you work out the intervals, not with respect to the tonic, but between each note, you'll notice that the interval between the 6th and 7th degrees of the harmonic minor scale is pretty large.
It is an 'augmented major 2nd' - C to D# [not a minor 3rd – C to Eb]
You can also call this interval a "sharpened major 2nd"
and this just happens to 'sound' just like a minor 3rd. This is all well and good for us musicians but there was a problem, choirs hated singing this interval especially with melodies in the ascending direction. It really is pretty tricky to visualise and then sing this interval.
Also, this large interval is particularly 'smooth' when it comes to melody and 'ornamentation' [things like trills].

The solution was to reduce the size of this interval by raising the minor 6th of the scale to a major 6th [during the perfect cadence over chord V7]

Now we have yet another newly modified scale spelt:
Tonic
major 2nd
minor 3rd
perfect 4th
perfect 5th
major 6th <- our new modified note
major 7th
this scale is called the 'melodic minor ascending' scale, and sounds very cool too.

So using E as the tonic we get: E, F#, G, A, B, C#, D# [used over the chord of B7 [V7] in the key of Em]

However, in the descending direction, the 'melodic minor descending' scale is the same as the natural [relative] minor.

For these scales and the perfect cadence, this really is the tip of a huge and incredibly useful / interesting iceberg.

For now, knowing that these modified scales exist and why is good enough, if for no other reason than to de-mystify the 'what' and the 'why'.
In the early stages of playing and learning the neck, concentrate more on the regular major and minor scales.
As you become a more advanced player I'm sure that you'll begin to get into using these scales in anger.

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Frank