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 honour alongside it. Often warriors who loses in battle would have their meito taken from them by the victor, thus losing their honour alongside it as to lose a meito is to disrespect the honour 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 honourably 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 which is often recorded by fans as "Meito Shusui" instead of just "Shusui" in error by those who do not understand that Meitou is a status) or as a "renowned blade" (understandable given that if you break up 名刀 it forms 名 for renowned and 刀 for sword.)

Tashigi's dream is to free all the meito from the hands of villains 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 bad person. As of Chapter 541, she has no victories over Zoro in her quest and as far as we 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 aquired 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 "Kitestsu" 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.
 * 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 "Kitestsu" 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.

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.