Japanese honorifics are traditional suffixes added to a person's name to convey a level of respect or familiarity with a person. For the most part, they do not translate well into English. Most fanfiction writers will come into contact with Japanese honorifics through fansubbed anime and scanlated manga. Some professional companies do leave them in, with glossaries to explain their meaning and usage, but this does not mean that the fanbrats that follow those series will be any more proficient in their use than others who don't.
Japanese honorifics are one of the few pieces of Japanese that are acceptable for use in English language fics, as long as they are used correctly. If you can't use them correctly, or are not sure of how a particular honorific is used, then you should leave them out. It is best to rewatch or reread the necessary parts of the canon to figure out what honorifics characters use, if you can't remember offhand. Incorrect use of honorifics is counted as a charge by many agents, although not one serious enough to be counted as cause for assassination or exorcism on its own.
It is also important to remember that some canons only use honorifics in their original version because they were originally produced in Japan. Characters such as Alucard from Hellsing and Duo Maxwell from Gundam Wing are not likely to use honorifics in everyday speech. Having a character who is not Japanese in canon use honorifics may also be counted as a charge.
The most commonly encountered honorifics in fanfiction are:
San
San is a common suffix to add to a last name, basically the equivalent of Mr. or Mrs. or Ms. It is the all-purpose honorific and can be used in any situation where politeness is required.
Sama
This one is one level higher than "-san". The person being referred to as "-sama" is highly respected.
This is usually not appropriately used in fanfic.
Dono
This comes from the word "tono", which means "lord". While also a respectful term, it is an even higher level than "-sama" and confers utmost respect. It also has a more political connotation.
Kun
Kun is primarily used at the end of the boys' names to express familiarity or endearment. While it is not appropriate to be used to call people older than yourself, it can be used for boys younger than and of your age.
Chan
This is used to express endearment, mostly towards girls. It is also used for little boys, pets and even among lovers. It gives a sense of childish cuteness. However, calling a stranger "-chan" can be considered to be disrespectful and rude.
Tan
Tan is a babytalk version of chan, used primarily by small children and people trying to be cutesy.
Calling strangers "-tan" is also disrespectful and inappropriate.
Bozu
This is an informal way to refer to a boy, similar to the English terms "kid" and "squirt". It is similar to "gaki".
Sempai or Senpai
This title suggests that the addressee is one's senior in a group or organization. It is most often used in a school setting, where underclassmen refer to the upperclassmen as "sempai". It can also be used in the workplace, such as when a newer employee addresses an employee who has seniority in the company.
Kohai
This is the opposite of "sempai" and is used toward underclassmen in school or newcomer in the workplace. It connotes that the addressee is of a lower station.
Sensei
Literally meaning "one who has come before," this title is used for teachers, doctors, or masters of any profession or art.
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This is usually forgotten in these lists, but it is perhaps the most significant difference between Japanese and English. The lack of honorific means that the speaker has the permission to address in a very intimate way. Usually, only family, spouses, or very close friends have this kind of permission. Known as yobisute, it can be gratifying when someone who has earned the intimacy uses it. When the intimacy hasn't been earned, it can be very insulting.