All scales in the diatonic/chromatic are constructed from a pattern of intervals. If you're not aware, our Western system of harmony consists of a chromatic scale with 12 tones:
C - Db(C#) - D - Eb(D#) - E - F - F#(Gb) - G - Ab(G#) - A - Bb(A#) - B
The distance between these consecutive tones is known as a half step (for example, the distance from C to Db). When we use interval patterns to construct scales, the easiest unit to measure the distance between notes is by counting the number of half-steps.
With that in mind, the three minor scales - Natural Minor (Aeolian), Melodic Minor & Harmonic Minor - can be viewed as an interval pattern like the picture
In this example I used A Minor as our reference because in its natural form it has no sharps or flats. For those who can't read music, the tone names are written above the staff. If you look below the staff, you'll see a number between each note. That is the interval pattern needed to construct that scale:
NATURAL MINOR: 2, 1, 2, 2, 1, 2, 2
HARMONIC MINOR: 2, 1, 2, 2, 1, 3, 1
MELODIC MINOR: 2, 1, 2, 2, 2, 2, 1
These patterns remain consistent regardless of key. You can start on any note, and so long as you apply this interval pattern, you will have assembled this scale.
RULES OF THUMB FOR IDENTIFYING THE SCALES
You may find that using the interval patterns is not the simplest way for you to remember the scales themselves. Here are a few tricks you can use on each scale to help you remember which is which:
Natural Minor - Natural Minor is what we call a mode of the Major Scale - specifically the 6th mode, or Aeolian. If you take any major scale, start on its 6th degree and play it up to the next 6th degree, you will have played a natural minor scale. Example:
You will notice that the note it starts on is 3 half-steps down from the scale's starting point (distance between A and C). Therefore, one simple way to figure out a natural minor scale from a major scale is to count 3 half-steps down from its starting note, or simply find the 6th note and play the major scale from there as shown in the picture.
Harmonic Minor - Harmonic Minor is the only minor scale with an interval larger than 1 whole step (two half-steps). One sure way to identify this scale is to listen for the leap between the 6th and 7th tones. The nature of that leap provides the scale with a sound that is often described as arabesque, middle-eastern, etc... This is worth noting, as harmonic minor and its modes have several names & purposes in other cultures -- especially Klezmer, the music of Eastern Europe, & India to name a few. Give the scale a listen and notice that discrepancy that sets it apart from the others.
Melodic Minor - This one is arguably the simplest. Melodic Minor is simply a Major Scale with a lowered 3rd. If you're in the key of C Major, for example, and you lower the 3rd degree of the scale (the E => Eb), you will be in C Melodic Minor - the parallel melodic minor scale of C Major.
A NOTE ABOUT THE EXISTING DISCREPANCY ON THE DESCENT OF MELODIC & HARMONIC MINOR SCALES
In reference to melodic & harmonic minor scales, there are two commonly accepted "spellings" for the scales. The more traditional, which stems from the classical tradition, is to 'resolve' these two scales by playing the natural minor on the descent. The modern tradition, as is popular among jazz artists and some contemporary composers, is to play the same tones ascending and descending. The diagram below highlights these two concepts visually:
Melodic and Harmonic Scales - Jazz and Traditional
In this day and age, it is important to be aware that this discrepancy exists and be comfortable with performing them either way. In essence, there are two separate and unique ways to perform each of these scales. In an audition situation, if asked to play these scales, make sure you clarify the preferred method of performance - Traditional (resolve on descent) or Modern/Jazz (same tones on descent).