Daimyô (Japanese: 大名) literally means 'big name'. It generally refers to regional military lords who were able to excercise de facto military and administrative control of an area.
During the Muromachi Period, the breakdown of centralized authority left the shugo with little of their original power. This power vacuum was exploited by ambitious families, who took the reigns of power into their own hands. Conflict between the daimyo erupted in the late 15th century, with some of the bloodiest fighting during the Ônin War, ushering in the Sengoku Period.
In that period, a daimyo was a warlord who ruled over a large area.
Eventually, the role of the daimyô was solidified and incorporated into the official government structure as the lands once again came under a strong centralized authority in the Edo Period.
In the Edo Period the term daimyo was standardized: it referred to a direct retainer of the shogun whose han (fief) was valued at least 10,000 koku. There were cases where a retainer of a daimyo had a han of over 10,000 koku, but they were not considered daimyo.
Tokugawa Ieyasu divided the daimyo into two groups depending on their relationship to him at the time of the Battle of Sekigahara in 1600. Those who were already his vassals at the time of the battle were considered fudai daimyô 譜代大名, (vassal daimyo). All others were tozama daimyo 外様大名, or "outside lords." The tozama are often said to have been exclusively those who sided with Ieyasu's enemies at Sekigahara, but this is a misconception, and is strictly speaking incorrect. The tozama in fact included both the enemies of the Tokugawa, and those who were neutral in the Tokugawa-Ishida conflict, as well as the Tokugawa's most powerful allies; the latter were powerful enough to have not been subordinate to the Tokugawa in 1600, and should instead be seen as having been allies on a more or less equal basis with the Tokugawa.
These classes of daimyô were fixed for the duration of the Edo period; the shogunate altered daimyô ranking and territory at times, but daimyô were never shifted from one daimyô category to another. Ieyasu also set up a third class of daimyô consisting of his descendants, the shinpan daimyô 親藩大名, or "collateral daimyô."
Fudai daimyo were, with a few exceptions, vassals of Ieyasu before 1600 and their descendants. They included those who became daimyo during the Edo period, mostly bureaucrats whose stipend was raised to 10,000 koku. Relatives of Ieyasu, most of whom were allowed to use his original name of Matsudaira, were also included in this class.
The fudai daimyo, especially the lesser ones, and the hereditary vassals below daimyo rank were the bureaucracy of the shogunate. For many of the bureaucrats, their "han" were scattered pieces of land whose total income made up the required amount.
Tozama daimyo were daimyo who had not been vassals of Ieyasu in 1600 and their descendants. Many of them, especially the greater ones, had close ties with the shogunate, including marriage ties, but they did not take part in the bureaucracy or concern themselves with national affairs, at least publicly. When in Edo, they presented themselves at court on stated occasions and sometimes were given special duties. Thus they could devote themselves to the affairs of their fief for the most part, even when in Edo. However, at the end of the Edo period the foreign threat drew them into national affairs, especially when Abe Masahiro started consulting them.
The tozama daimyo were usually the ones the shogunate called upon to carry out any difficult or expensive undertaking. They were not taxed as such, though. During the early Edo period the shogunate placed the tozama daimyo under very tight regulation and took any excuse they could to confiscate their fiefs. However, in the Mid Edo period it was decided that the social problems caused by confiscation, such as the creation of ronin, outweighed the danger of revolt by daimyo, so various restrictions were modified.
The tozama daimyo were divided into two groups, those who had come into prominence under Oda Nobunaga or Toyotomi Hideyoshi, shokuhô 織豊 daimyo such as Maeda, Hosokawa, Kuroda, Asano, Yamanouchi, Sengoku, etc., and the "Old families" kyûzoku 旧族 who had been prominent before, such as Date, Shimazu, Môri, Uesugi, Nabeshima, Sô, etc. 
These were descendants of Ieyasu. They were in theory advisors to the shogun, but they did not have a place in the bureaucracy. One main purpose was to provide an heir to the shogunate if necessary.
The Three Lords (sankyô 三卿), Tayasu, Hitotsubashi, and Shimizu, were descended from two sons and a grandson of the shogun Tokugawa Yoshimune.
Others, called "Within the Gate" (kamon 家門）were descendants of sons of the first three shoguns and branches of the Three Houses, such as the daimyo of Fukui (Echien), Mastue, Saijô (in Iyo province), Aizu, and Takamatsu.
When the main shogunal house lacked an heir, one was to be chosen from the Three Houses or from the Three Lords. This happened three times. When they became shogun, Yoshimune and Iemochi were the daimyo of Kii, and Yoshinobu was head of the Hitotsubashi house. (Yoshinobu was originally of the Mito clan, but had been adopted as heir of the Hitotsubashi house.)
The heads of the Three Houses and the Three Lords and their heirs were allowed to use the name "Tokugawa." Shinpan daimyo of other houses and younger sons of the heads of the Three Houses used the name "Matsudaira."
- An example is the Andô 安藤 family who were retainers of the Kii Tokugawa clan. They ruled the 38,000 koku han of Tanabe in Kii province.
- "Fudai" first meant generation after generation or a family tree, then someone one serving a lord generation after generation.
- Of course, "descendants" in the Edo period meant descendants in the male line, including adopted heirs. Adoption of close relatives was preferred though, so even adopted heirs were often descendants of the same person.
- Hideyoshi had divided daimyo into fudai and tozama shortly before his death. "The fudai were those vassals who entered Toyotomi service young, voluntarily, without large holdings and during the early phases of Hideyoshi's career. The tozama, daimyo with independent land bases, submitted to the Toyotomi following alliance, negotiation, or defeat." (Mary Berry, Hideyoshi, Harvard University Press, 1982, p. 68) Though I have not seen Hideyoshi's list, his fudai and tozama daimyo probably corresponded closely to the Edo Period Shokuhô/Kyûzoku distinction.
- Sansom, George. A History of Japan 13334-1615. Stanford: Stanford University Press, 1963.
- Hall, John Whitney. Government and Local Power in Japan 500 to 1700: A Study Based on Bizen Province". Princeton: Princeton University Press, 1966.
- Hall, John Whitney, and Toyota Takeshi. Japan in the Muromachi Age. Los Angeles: University of California Press, 1977.
- Fairbank, John K., Edwin Reischauer, and Albert Craig, East Asia: The Modern Transformation, Boston: Houghton Mifflin, 1965.
- Kôjien Dictionary
- Reischauer, Edwin, and Albert Craig, East Asia: The Great Tradition, Boston: Houghton Mifflin, 1960.
- Totman, Conrad D., Politics in the Tokugawa Bakufu 1600-1843, Harvard Universiy Press, 1967.
- Mark Ravina, "Political economy in Tokugawa Japan: are tozama and fudai meaningful categories?," Clioviz (blog), 16 Dec 2012.