Many people mistakenly thing that the sword the Japanese Samurai used in combat was the Katana. The Katana is actually the youngest of the three main swords used in Japan and gained its popularity during the Edo period, aka, the approx. 300 years of peace. Here are the three main swords with descriptions of each. And yes, in the Bujinkan we study all three, because each one is used differently. Be aware, each sword also has it's own sub-variations, these are the primary categories of Japanese swords.
The Tachi was the sword used by Samurai during the Sengoku period, the era of the warring states of Japan. This period was the height of the Samurai and their abilities. This was the true Samurai Sword. It is single edged and has a significant curve to it. The handle is also curved and can be used with one or two hands. This sword hangs loose at the warrior's side like a saber, blade edge down. It was designed to specifically defeat the armor worn by Samurai and foot soldiers. You can see the cord wound in the middle of the sheath (Saya), this was used to tie the sword around the waist of the samurai outside of the armor. It was ideal for battle field fighting and fighting from horseback.
The Katana was developed as a result of the Edo period. After the last civil war of Japan ended, the new Shogun established a period of peace that lasted for 300 years. The Samurai found the Tachi to be too cumbersome to wear daily, and so the Katana rose to popularity among the samurai. The Katana is less curved than the Tachi and is worn tucked into the belt (Obi), blade edge up. The cord was worn loosely between the sheath and belt, not wrapped around the waist. The handle was usually at least 12 inches long for two handed use. This was in essence, a dueling sword or self-protection sword, rather than a battlefield sword. A revolution in Japan during the early 1800s led to the abolishing of the Samurai class, and so the Katana was the last Samurai sword, and the one most remembered.