h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

The main thing about the Katana was the differential cooling.

The folding introduced multiple thin layers of impurities which acted as effective crack stoppers in the main body of the sword, but it was the rapid cooling of the edge that created its hard martensitic structure which could hold the sharpness.

This was done by coating the blade with clay before heating it for the final quench: thicker clay over most of the sword, shading to a thinner amount over the very edge. The limit of the martensite part of the blade can be seen as the wave-like pattern, which is highly prized, although I can't recall what it's called ATM...

Al:
The wavy temper line is called a "hamon", from a combination of "ha" (edge) and "mon" (symbol/emblem/pattern).
The hamon is created by the differential cooling of the edge in tempering - the clay coating means that the edge is cooled quickly, remaining very hard, while the back of the blade cools more slowly, becoming softer but tougher.
You are left with a blade with a very hard edge supported by a tough spine, assuming of course that the blade does not crack.
The differential cooling is also the cause of the sword's elegant curve.

Oh, and the repeated folding and hammering of the steel has the effect of removing a lot of the impurities, which would otherwise cause discontinuities in the crystal structure of the steel (potential points of failure). The advantage of damascus or pattern-welded steels is that the technique reduces the unevenness in the poor quality steel which was available.

As an aside, the "folded 2000 times" myth probably arises from the number of layers - every time you fold, you double the number of layers, hence 2, 4, 8, 16, 32, 64, 128, 256 etc. Folding only 9 times gives you over 2000 layers, which is far more reasonable - the folding and hammer-welding process takes a lot of time, materials and skill, and the law of diminishing returns applies.