One of the three types of minor scales. This scale (along with the harmonic minor scale is used heavily by heavy metal guitarists such as Yngwie Malmsteen.

To form the scale, we use the Harmonic Minor and simply raise the 6th degree 1/2 step. C harmonic minor:

C D Eb F G Ab B C |___|___|____|___|___|____|___| 1 1/2 1 1 1/2 1.5 1/2

The numbers between the notes denote the interval between them. Now, to derive our melodic minor, we simply raise the VI degree of the harmonic minor scale (Ab -> A). Here we have it:

C D Eb F G A B C |___|___|____|___|___|___|___| 1 1/2 1 1 1 1 1/2

Voila!
A note on improvisation: Historically, this scale has been played the way it is written above only while ascending the scale. This is because of a tendency in classical music to make runs up and down the scales to define tonality. However, if you come back down the melodic minor scale, it starts to sound like a Major scale. Since we want the minor tonality defined right away, classical musicians played the natural minor scale descending in the melodic minor scale. However, since most of its usage in rock has little to do with classical music, this convention has become a bit obsolete. If you plan to use this scale to improvise in rock (hey, why not), I suggest you play the same scale up and down. If you're a classical nut, don't let me stop you from playing the Aeolian down. After all, it's your ]]life.
Basically, the scale is used over any kind of minor chord to solo.

If you're familiar with II-V-I progressions, you can use your knowledge to create new ones using the harmonic minor scale as the base scale. Simply slap in your notes, and form the chords on top of that.

The melodic minor scale contains seven modes. They are, from the first up (in the key of A minor):

The guitar pattern for the scale is:

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The bass pattern for the scale is:

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The piano pattern for the scale is:

A  B  C  D  E  F# G# A
1  2  3  1  2  3  4  5

Source: The Keyboard Grimoire (book), Advanced Scale Concepts and Licks for Guitar (book)

The Melodic Minor - The WHY?!:

Well by now you might wonder why on Earth this whole ascending and descending business is going on (refer to the writeups above about how the melodic minor scale changes depending on if it's ascending or descending). It's the only scale used commonly to do such a whack thing and the reason is because people can't sing in key.

Let me explain.

Background. Back in the day, that being the 9th century AD or so, monks used to sing Gregorian chant, Anglican chant, and Plainsong in the monestaries in Europe. These songs had some semblances of rhythms, including a few rhythmic modes, but no meter as we understand it today. They also didn't have chords or harmony. Eventually, these chants were worked modes, based from Greek modes: Aeolian, MixoLydian, Lydian, HypoLydian, HyperPhrygian and HyperLydian. But I digress.
You and I take the idea for granted, but chords weren't really used at all back then, and when monks started getting spunky and singing a fourth away from the melody line, thereby implying chords, the concept of "tonic" was born. Also, the first Church Modes were born through music theory.
Tonic, in a musical sense, is the "most important note" in a song or key. It's what a song is "in" when I say that Mozart wrote something "in" B-flat. 4,999 out of 5,000 songs you've heard end on a tonic chord.

The Reason. Well, since chords and this tonic thing being a new to those monks, they had a tendancy to go sharp when they approached tonic at the top of a scale. When they descended the scale, they'd sing in tune again (being drawn to the dominant, fifth note of the scale). Sharps and flats had not yet been "invented" (or at least acknowledged) yet, and so "accidentals" - that being sharps, flats, naturals, double-flats, and double-sharps - were added to account for their errors. Today we still use the scale.

Reasons that are less amusing.
The melodic minor is able to have a leading tone in the melody by raising the 7th. This gives a nice 'push' up to the top note, tonic. Still, since you don't want an Augmented 2nd (1 1/2 steps) in the scale (which sounds Oriental), you raise the 6th as well. When descending the scale, it is not necessary and usually unwanted to have a half-step, because of it's strong tendency to go up.
So this is the scale used in the melody (hence the name) - it is modified depending on a scale. That is why you have the different forms of minor scales. Natural minor is the base form, and the harmonic tweaks with the 7th so that the V and vii chords are Major and diminished (instead of minor and minor, which would be unusual and strange to Western ears). Often, a voice (tenor, bass, soprano, alto, an instrument, whatever) will change minor modes depending on the situation. Still ... I can just picture those monks ... warbling .....


Reality check.
Really, though, the reason we have these three minor scales (Melodic, Harmonic, and Natural) is because people decided that way they sounded good and used it in their music. When a pattern is used frequently in real music, it is useful to pin it down as 'theory' and practice it as an exercise.


Sources:
http://janpress.freeservers.com/LCscales.htm
http://home.datacomm.ch/straub/mamuth/mamufaq.html#Q_melodicminor
My old music theory teacher, Larry Solomon http://solo1.home.mindspring.com/

Thanks to Footprints for making me put a bit of actual music theory into my writeup.
Any corrections are welcome - the exact chronology of the evolution of modes in relation to chords is something I'm still trying to figure out.

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