Here's something I've saved from the past on the subject. It's quite a bit and has a lot of history, but it has a point:
Explaining the Harmonic and Melodic Minor Scales
The whole thing with minor scales is a complicated one, but here's a brief history/understanding of why both of you are right. It started back in the Renaissance when theories of harmony were first being developed out of polyphony using the modes. Composers found that the ionian mode (major scale) gave them the most options when creating pleasing harmonic progressions because of the leading tone, or natural (major) seventh degree of the scale. The V-I cadence that we all know and love came from this. They liked the sound of the natural seventh going up to the tonic so much that they started briefly changing notes of the other modes to reflect the natural seven going to one in something they called musica ficta, or a chromatic note. They weren't part of the scale, but they were used because they created a pleasing sound (the natural seventh resolving to the tonic). For whatever reason, as the years went by, the only mode that they kept using with the musica ficta was the aeolian mode, which you may know as the natural minor. However, once you made the lowered seventh a natural seventh, you have the harmonic minor.
Get it? The natural seventh was raised for the pleasing harmonic cadence of V-I (V major, not V minor, which you would get with pure aeolian), so thats how you get harmonic minor. The reason for melodic minor comes from this lineage. If you raise the seventh degree of the minor scale, you get an odd interval between the lowered sixth and the seventh. For example, if I were to spell A harmonic minor....
A B C D E F G# A
...the interval between F and G# is an augmented second. This is a very unusual and awkward interval for singing, and since most music was vocal at the time that these theories were being developed, the augmented second was heavily eschewed. Even though its enharmonic to a minor third, in context the sound of the augmented second has quite a different character. So, in order to keep the important natural seventh (G# in the key of A minor), you would raise the sixth degree to smooth out the linear melodic contour, and then you have melodic minor. In practice, composers used all three minors simultaneously in what theorists have called the "composite" minor.
This is all well and good, and its a nice classical theory lesson for you, but how does it apply to playing contemporary music? Well, this is why you're both right. All music which is functionally tonal comes from the classical tradition. Functionally tonal means that the chords within a chord progression all tend to lead to a V to I cadence. This means if you're playing any music that has a V chord going to a I chord, it's tonal, and its functional. These are the circumstances where you would more likely than not find the melodic minor. Since the natural minor lacks the natural 7 that is so vital for the V chord, melodic minor shows up as a more "true" minor scale. However, if you're playing music which is not "functionally tonal", like music that just has one chord, or if you're playing music based upon a riff, or if you're playing music that has a chord progression, but no V-I cadence or any kind of V major chord in it, than natural minor is more of the "true" minor scale. What it comes down to is the V chord. If you're playing music with a V7 or V major, think melodic minor.
Here, to make it clear, do this.
Harmonize each scale. You do know what "harmonize" means, right? OK, if not, it means to stack thirds from the scale on top of each other to build four-note chords.
Using A natural minor you get:
Amin7, Bmin7b5, Cmaj7, Dmin7, Emin7, Fmaj7, and G7. Don't take my word for it, build 'em yourself and check my work!
That G7 pulls your hear to the Cmaj, not the Amin7. So, as haVIC5 says, another scale was made to fix that. If you raise the G to G#, you change every chord that has a G in it... Now you get
Amin/Maj7, Bmin7b5, Caug, Dmin7, E7, Fmaj7, G#dim. The G7 is obliterated so it doesn't pull to the C chord. And the E7 sets up a strong pull to the Amin7. That makes the harmony work much better for most people's ears. That's why its called the "Harmonic Minor"- these names don't generally come out of nowhere! However, that scale sound weird (of course "weird" is relative- where would Richie Blackmore be without the Harmonic minor?!). The minor 3rd (technically augmented 2nd as HavIC5 explains very neatly) is weird, they made changes to the Harmonic minor.
By raising the 6 (F) a half-step, you've smoothed out the melodic problems with the Hamronic minor. And since the problem with the meolody was generally associated with ascending, the classic melodic minor uses F# ascending and F natural descending.
Melodic Minor != Jazz minor. Melodic minor in the classical sense has an altered 6th and 7th degree when ascending and a natural 6th and 7th when descending. A Jazz Melodic Minor has an altered 6th and 7th both ascending and descending.
Harmonic Minor is the natural minor scale with a raised 7th scale degree.
Natural Minor != Relative Minor. That right there is a perfect case of apples and oranges. C natural minor is not the relative key to C major.
Listen to Richie Blackmore who'd be DEAD without a Harmonic minor (well, not really, but maybe his more annoying protege, YJM would be 
).
