Meito

A Meito (名刀) is not a type of sword but rather the status of a sword.

In Japan, this means the sword has a history and is of legendary status. The blade is normally a quality blade which stands out above all other swords in its endurance and sharpness.

A meito is sometimes associated with the warrior who wields it and becomes part of his identity. Regardless of who owns them, a meito is regarded with highest respect in Japan and to be given a meito is to carry a great honor alongside it. Often warriors who lose in battle would have their meito taken from them by the victor, thus losing their honor alongside it as to lose a meito is to disrespect the honor of wielding it.

To wield a meito honorably, usually it would be passed down to them from master to pupil or from another family member. There exists several other methods alongside this which are acceptable but uncommon. Another method of gaining a meito honorably was to be offered it, this was usually out as show of respect from one person to another. A warrior may also win it in battle against another warrior in a fair duel and thus be allowed to claim it as their own.

To steal a meito or gain it using another dishonorable action is to show the meito disrespect and warriors who do so were often frowned upon. To put the status of a meito in a western style of thinking, a meito would be considered in status on par with swords like Excalibur (which would be regarded as King Arthur's "meito" in the Japanese sense).

One Piece
In One Piece, meito is often mistranslated as either part of the name of a sword (e.g. Zoro's Shusui is often recorded by fans as "Meito Shusui" instead of just "Shusui", ignoring the fact that meito is a status) or as a "renowned blade" (understandable given that if broken up, 名刀 gives 名 "renowned" and 刀 "sword").

Tashigi's dream is to free all the meito from the hands of criminals such as pirates. She believes the swords are crying because instead of being used for noble and honorable deeds, they are being used for criminal use. Her reason for tracking Zoro is to gain ownership of his meito, the Wado Ichimonji, which she believes is in the hands of a malicious person. As of Chapter 541, she has no victories over Zoro in her quest and, as far as readers know, while she has taken the sword Kashū - a sword not listed as a Meitou but still one of the Ryo Wazamono Grade Swords - she has taken none of the meito in her journey of Justice, while Zoro still possesses Wado Ichimonji and has also acquired Shusui.

Meito is translated simply as Famous Sword in the Viz Manga and Legendary Sword in the FUNimation dub.

12 Saijo O Wazamono Grade Swords

 * Kokuto Yoru is one of the twelve Saijo O Wazamono, under the ownership of Dracule Mihawk.
 * Shodai Kitetsu is the first sword of the "Kitetsu" series. It has yet to appear.

21 O Wazamono Grade Swords

 * Wado Ichimonji was Kuina's family meito and Zoro was allowed permission to wield it as part of their promise to each other and carry with it her dreams.
 * Shusui was Ryuma's meito which belongs to Zoro now. Upon defeat, Brook's zombie offered it to Zoro as a sign of honor for their battle.
 * Nidai Kitetsu is the second sword of the "Kitetsu" series. It has yet to appear.

50 Ryo Wazamono Grade Swords

 * Yubashiri was Ippon-Matsu's family meito and he gave it to Zoro after seeing him testing the cursed Sandai Kitetsu, knowing the sword would have a better master than staying at the shop. It was destroyed during Zoro's fight with Shuu of the Marines, and its remains rest near the Rumbar Pirates' grave in Thriller Bark..
 * Kashu was a sword that once belonged to Baroque Works agent Mr. 11. Tashigi took it for herself after defeating Mr. 11.
 * Yamaoroshi was wielded by Billy the Orca Killer until he was beaten by Tashigi, who took his sword.

Wazamono Grade Swords

 * Sandai Kitetsu is the third sword of the "Kitetsu" series. It's wielded by Zoro, who received it as a gift by Ippon-Matsu.
 * Shigure is Tashigi's meito. Little else is known about this katana. It was destroyed when Tashigi tried to attack Trafalgar Law.

Unknown Grade

 * Oto and Kogarashi are a pair of double-edged swords which belong to Shiki.

Trivia

 * Though there are many named swords in the One Piece world, many of which fall into the katana grades, only a few of them have been mentioned so far as "meito" when introduced.
 * So far in the series, Zoro has held the most meito to date.