Meito

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

To wield a Meitou honourably, usually it would be passed down to them from master to pupil or from another family member. There exist several other methods alongside this which are acceptable but uncommon. Another method of gaining a Meitou 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 Meitou or gain it using another dishonorable action is to show the Meitou disrespect and warriors who do so were often frowned upon. To put the status of a Meitou in a western style of thinking, a Meitou to would be considered in status on par with swords like Excalibur.

One Piece
In One Piece, Meitou is often mistranslated as either part of the name of a sword(e.g. Zoro's Shuusui which is often recorded by fans as "Meitou Shuusui" instead of just "Shuusui" in error by those who do not understand that Meitou is a status) or as a "named blade"(understandable given that if you break up 名刀 it forms 名 for name and 刀 for sword.) The latter is more accurate but still incorrect as any blade can have a name, but only a few swords are Meitou.

Tashigi's dream is to free all the Meitou 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 Meitou, the Wadō 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, none of the other Meitou in her journey of Justice.

List of Meitou

 * Wadō Ichimonji was Kuina's family Meitou and Zoro was allowed permission to wield it as part of their promise to each other and carry to with it her dreams. Zoro tends to keep this sword in his mouth and apply it for his steel cutting techniques besides his frequent use of it in Santouryu situations where he is forced to use one sword. It is the only sword that Zoro has had from the beginning of the series and never got broken.
 * Shigure is Tashigi's Meitou. Little else is known about this katana, but she seems to be very proficient wielding it.
 * Yubashiri was Ippon-Matsu's family Meitou, 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. The sword was destroyed by Shuu, and is now laying at rest at Thriller Bark.
 * Ryuuma's Meitou was his sword Shuusui which belongs to Zoro now. Upon defeat, Brook's shadow offered it to Zoro as a sign of honor for their battle. Another reason why he gave the sword to Zoro was because he shamed the Samurai it inhabited by losing to Zoro. Zoro compared it to "a child" yet he was able to hold back Oars alone with the blade. This strongly implies that he isn't capable of using its full power yet.

Trivia

 * Though there are many named swords in the One Piece world, only a few of them have been mentioned so far as "Meitou".