A Japanese phrase, meaning literally "kindly", "acceptably", but in practice meaning just about anything the speaker wants to say but doesn't have the guts or doesn't feel the need to say. Understanding this word is key to understanding the Japanese culture.
"Sachiyo-san ni yoroshiku." "Say 'yoroshiku' to Sachiyo for me." = "I kinda like Sachiyo but I'm afraid to ask her out, so please tell her I said this so she can wonder if maybe I like her and then hopefully communicate via subtle body language at some point that she might like me too, or not."
"Kongo mo douzo yoroshiku onegai shimasu." "'Yoroshiku' in the future, please." = "I hope to at some point in the future get your business, but I don't have the balls to ask for it directly. Perhaps if I call on you twenty more times over the next three years, you'll be impressed with my sincerity and persistence and buy some of our product."
"Hakata-kun, ano repooto yoroshiku ne." "Hey Hakata, 'yoroshiku' on that report" = "I need you to do that report for me but I don't want other people to think I'm weak by actually asking you to do it, so I'll just not-so-subtly hint that I want it done so they can see my dominance as I get what I want without even having to ask for it."
Yoroshiku is a phrase used only in Japanese, which is what you usually say to somebody as a greeting. It's difficult to translate to English, but it is used as a polite "Nice to see/meet you." and is usually accompanied by Hajimemashite (Nice to meet you.)
Person 1: "Hajimemashite, Person 1 desu. Yoroshiku." (Translation: Nice to meet you, I am Person 1.)
Person 2: "Hajimemashite, Person 2 desu."
(Translation: Nice to meet you too, I'm Person 2.)