A modern Japanese name (人名 Jinmei) is made up of a family name, or myoji (苗字 or 名字, myōji), followed by a given name, which can be called the "name" (名前 namae) or "given name" (下の名前 shita no namae). A suffix title such as "-san" (Roughly "Mr." or "Mrs.") is used after the name.
A typical Japanese name is Yamada Taro (山田太郎 Yamada Tarō - Yamada is the family name, Taro the given name). Taro's son might be named Yamada Ken. In other words, Taro and Ken are two members of the Yamada family.
In Japan, a married couple must match their surnames -- this is a legal requirement -- and in most cases, the wife adopts her husband's surname. But some women who have professional career continue to use born surnames as their personal 'brands' even after marriage. For example, Umezawa Yukari married to a footballer whose surname was Yoshihara in 2002, thus became Yoshihara Yukari legally. She continued to use the born surname Umezawa as a Go pro. In 2011, she finally changed her professional name (that is, registered name at Nihon-Kiin) to Yoshihara Yukari.
Note: In China and Korea, married women keep their born surnames.
There are many issues regarding Japanese names in English, including naming order and romanization.
Japanese names are "switched" to the Western order (given name, then family name) when many modern-day Japanese individuals are discussed. E.G. Koizumi Junichiro (小泉純一郎 Koizumi Jun'ichirō) becomes "Junichiro Koizumi" in English.
Japanese names are usually kept in the original order when historical figures or figures in traditional Japanese activities (such as Go or Waka) are discussed. For this reason, Sensei's Library uses the traditional Japanese naming order (family name, then given name) almost universally; Go players are known in the traditional Japanese order by the Anglophone Go community, so Sensei's Library will use the original Japanese naming order when discussing Japanese Go players.
Due to the choices of the United States publisher of the anime and manga series Hikaru no Go, Sensei's Library will use the western order (given name, then family name) when discussing most Hikaru no Go characters (those characters from modern day Japan).
Referring to a Japanese person by the given name alone presumes a certain personal familiarity and might be rude in some contexts. Except in situations of some intimacy Japanese people are referred to by the family name. Consequently if you aren't sure which name is the family name it might be best to use the whole name to avoid confusion.