yoroshiku onegai shimasu 

Basically, this is said as an entry-level phrase to someone you have a favorable impression with, or if you are a very 'traditional' person. Another situation you would use this, is if you were being convinced to accept someone's hospitality. I am unfamiliar with the direct transliteration, but if you wash the grammar properly it comes out to be 'Please treat me kindly' or 'Please take care of me' or 'I am in your care, kindly' something to that effect. As you can see, this phrase is a trade of trust, where you 'ask' for trust, and 'give' trust by 'placing' yourself in someone's care.

Source? My frequent visits to Japan, and familiarity with its culture.

'Yoroshiku onegai shimasu, senpai.'

- Here, an underclassman at school or work who might have previously met this person they refer to as 'senpai' (or senior, upperclassman) Implores them to continue their goodwill now, and in the future. The meeting may be in front of the senior's friends, so the person wants to be respectful so their senior doesn't lose face.

yoroshiku onegai shimasu by Taketo November 26, 2013

Yoroshiku 

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.)

Yoroshiku by Colin E. July 9, 2006

Yoroshiku 

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 by aaronsama September 26, 2011