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1.
A japanese anime which has been subtitled in English by fans and released by one of many fansubbing "groups". Typically these are series which have not been licensed for release in an English format, and hence releasing them is unlikely to result in legal action (although it is still technically illegal under international law.)

Typically, once a series becomes licensed for release in an English format, many fansubbing groups will cease subtitling of that anime, for fear of legal action. However, some rogue groups will subtitle licensed anime.
The latest Naruto fansub has just been released by ANBU.
by polarity March 21, 2004
 
2.
abb. for fan subtitles, a copy of a foreign movie, tv show , anime etc. in which subtitles have been added in the viewers native language
i waited all week for the fansubs version of initial d , my japanese ain't that good.
by defsix June 18, 2006