The so-called "natural" minor scale can be thought of as the base minor scale. If we were to play from A to A on the piano using only the white keys we would be playing an A natural minor scale. Sometimes this is called the Aeolian mode.
Now, you may notice how the natural minor scale lacks a leading tone. Instead, between the 7th scale degree and tonic there exists a whole step. Now, that's fine, but it could be better. We can really increase that tension and magnetic pull from the leading tone (scale degree 7) up to the tonic by raising that pitch up by a half step. Think of the sound from "ti" to "do," or play G# to A on the piano; this has a stronger pull up to tonic, A, than "te" "do" or G to A. Looking a bit deeper we can see that raising that leading tone enables us to have a major harmony built on the 5th scale degree, which is the all-important Dominant. With the natural minor scale we would have EGB, which is an E minor chord. Raising that leading tone gives us EG#B, a nice solid major dominant. This is why when we simply raise the leading tone of the natural minor scale we call it the Harmonic minor scale, because we altered a pitch in favor of creating that dominant harmony.
If we were to play that harmonic minor scale, though, we might notice how from scale degree 6 to scale degree 7 (F to G#) we have what SOUNDS like a minor 3rd, but since it is a scale and they are neighboring pitches, is actually an augmented 2nd. That's kind of weird to have, so let's just go ahead and smooth out that kink by raising the 6th degree up to F#, which gets rid of any augmented intervals. Now F# to G# is a major 2nd. Problem solved. That's your melodic minor scale.