Little is known about Masamune's life. He apparently lived in Sagami province, near present-day Kamakura and was probably trained by swordsmiths from Bizen and Yamashiro provinces, present-day Okayama prefecture and Kyoto respectively. It is likely that he was trained specifically by Kunimitsu and Kunitsuna, two of the most respected swordsmiths of the day. Masamune also trained a number of important swordsmiths, including his son Sadamune. Approximately 50 swords by Masamune are believed to still exist, though not all that are attributed to him are actually signed by him.
In addition to the folding technique that jeremyf mentions above, these swordsmiths employed at least one other method of boosting quality. Iron with different levels of carbon content were forged together--higher carbon content for the sharp cutting edge, lower carbon for resilience and flexibility. This particular technique reached its peak during Masamune's time.
While these techniques led to a high quality sword, they are not exactly why Masamune and his school are today considered artists. Samurai of the time period wanted a weapon to be both functional and beautiful, so swordsmiths attempted to add beauty to the sword itself. Masamune and his school did this in two ways. The first method was carving. The carvings appear more often on the shorter swords and are typically derived from the rules and prescriptions of Esoteric Buddhism.
The other technique is known as hamon. Hamon, or temper patterns, are possibly the most beautiful aspect of Masamune's work, and the one that he employed more skillfully than his teachers or students. Basically the heated blade was coated with different thicknesses of clay, applied in patterns, then plunged into cold water. The varying times of cooling achieved thusly created contrasts in the color of the metal, leaving swirling, subtle patterns of light gray on darker gray on the blade. These are often compared to clouds, and indeed while you look at them, they appear to move. While the hamon of Masamune's teachers are fairly simple (except those of Yukimitsu), Masamune's hamon exhibit great depth and complexity, and most of his students seem to have followed this example.
Sources: Liddell, C.B., "Master at the Cutting Edge of Art", The Japan Times, May 29, 2002.
Masamune: A Genius Swordsmith and His Lineage, Catalog of the special exhibition: compiled and published by Sano Art Museum, Suiboku Museum, Tokugawa Art Museum, and Nezu Institute of Fine Arts; 2002. Curated by Takeo Watanabe. (nb: I saw the exhibit at the Nezu Institute in Tokyo on June 8, 2002.)