The Masamune has appeared in several Final Fantasy games, going all the way back to the first installment. In a number of these instances, such as in Final Fantasy V, the Masamune ranks among the most powerful of weapons. As it appears in Final Fantasy VII, the Masamune takes on a whole new signifigance. Instead of being merely another equippable sword, the Masamune is the signature weapon of the central villan, Sephiroth.
The Masamune takes its name from actual swords made by Goro Nyudo Masamune (1288 - 1328), who lived during Japan's Kamakura Era and is considered by many to be the finest Japanese swordsmith of all time. Masamune's blades are incredibly rare in this day and age, and are considered national treaures in Japan.
The reputation of Masamune's swords is such that they are associated with a warrior who is internally peaceful and calm, as opposed to the swords of Muramasa (another famous Japanese swordsmith), which were described as bloodthirsty and evil. There is a legend where a Masamune sword and a Muramasa one are placed in a stream. In the stream, Muramasa's sword cuts a floating leaf clean in half, while a second leaf abruptly changes course when coming near Masamune's sword, bypassing it altogether.
When playing FFVII, one may notice that when Sephiroth pierces Aeris with the Masamune, there is no blood. Despite the legend recounted above, there might have been practical concerns at work in the creation of this FMV—it's possible that Square wanted to avoid a "Mature" rating. Some visitors to the Citadel have noted that blood appears in other points of the game (Jenova's "escape" from the Shinra building and Sephiroth's defeat being two prime examples), but the (already) shocking nature of Aeris' murder scene is such that the addition of blood would've tested the boundaries of what is acceptable for Square's intended audience. It's also worth noting that splattering blood is also not exactly the easiest thing to animate, particularly for a studio which (at the time) was only just moving into the medium of 3D computer animation. It's worth pointing out that Tetsuya Nomura's character design for Sephiroth shows what looks like a large red stain on the Masamune's blade; evidently, within the game itself, any blood drawn by the sword is probably implied.
Other Japanese Swords in FFVII
Two other swords with Japanese names that make appearances in Final Fantasy VII are equippable weapons for Cloud: the Murasame and the Yoshiyuki.
Murasame's name comes from Nanso Satomi Hakkenden, a 106-volume novel by Kyokutei Bakin written during the Edo period. The Hakkenden is a very famous story in Japan, and its influence can be found in a number of modern Japanese media. In this novel, the Murasame is a sword used by Inuzuka Shino, one of the story's eight heroes.
As for Yoshiyuki, there have been a few swordsmiths by that name, but I haven't been able to find any that are very distinctive in Japanese history. In fact, Yoshiyuki seems to be a common name in Japan, so it would be difficult to link one particular person by that name to Cloud's sword in FFVII.
Originally "Meaning of the Masamune Blade". Fully revised by Reeve, November 2005.
Contributors: Aeris Valentine, Broken Encryption, Goro Nyudo Masamune, M. Waiz, and Chigouki Hyuga.
Additional references: Hakkenden - The Greatest Samurai Novel in Japan