A Japanese expression of good will
, this word is largely untranslatable
Said on its own, (usually in the context of just having been introduced to someone), it means something like "hey, lets be nice to each other". Usually you would be more polite than this, saying something like "yoroshiku onegai shimasu", as, if you just met the person, you don't know them well enough to use informal speech.
Technically, the word "yoroshiku" is just a more humble form of "yoku", meaning "well". The additional meaning associated with it is a cultural phenomenon. As an adverb, it can be used to modify a verb, generally with the sense that the verb is being conducted in such a way as to express good will.
This is complicating things though. This word is almost the verbal equivalent of smiling at someone. Contextually, if you say "yoroshiku" you're expressing that you trust someone, will be friendly towards them, are grateful to them, or just generally have nice happy feelings in their direction.
Note: the "i" is a whisper vowel, so the pronunciation is something like "yoh-roh-shku".