The Japanese phrase "Yoroshiku onegai shimasu" can have various meanings depending on the situation in which it is being used. When it is said during a first meeting, it means something to the effect of "Let us have a nice relationship" or "Please treat me well". It can also be said when asking for a favor however, giving it a meaning more like, "Please", "Make it so", or "I'm leaving it up to you".
Now, the standard "Yoroshiku onegai shimasu" is the polite version of this Japanese phrase. To make this phrase more formal, a person can say, "Douzo yoroshiku onegai shimasu", and for a more informal situation, this phrase can be shortened to "Douzo yoroshiku". The least formal version of this phrase is just simply "Yoroshiku".
I've made a short video to explain this, if you'd like to watch:
