This is one of the phrases that I have the most difficulty translating to another language. The first part, よろしく (yoroshiku), comes from the word 宜しい (yoroshii), meaning "good; OK; all right; fine; very well; will do; may; can". The second part, お願いします (onegaishimasu), meaning "please", comes from the word 願う (negau), meaning "to desire; to wish; to request; to beg; to hope; to implore".
The problem (for me) is when it's inserted into regular situations. It's hard to translate it in a way that it seems natural in the target languge (for those wondering, "target language" is the language you're translating to, "source language" is what you're translating from). In a way, it seems to me a pretty meaningless phrase, carrying no further meaning than to emphasize what has come before it, but it can also be used to imply a request.
Let's have a look at some translations of it.
This first example is from the drama-series "Great Teacher Onizuka". In this scene Onizuka first arrives at the school and introduces himself to the other teachers. The phrase is here translated as "I will be counting on your assistance, pleased to meet you." (Translation by Japan-TV Fansubs.)
This is Morning Musume appearing on the show Pop-Jam, with them meeting the person on the right for the first time, with him bowing, phrase translated as "Nice to meet you." (Translation by mpz.)
This is Morning Musume appearing on Utaban. They've appeared on the show multiple times in the past, so in this context it would be inappropriate to translate it as "nice to meet you", so this time it's translated as "Happy to be here", even though it's the same phrase in Japanese. You can also pay attention to how the older members (front line on the sides) don't bow as deep as the younger ones. (Translation again by mpz.)
From Morning Musume's new-years special, the spoken phrase is 今年もよろしくお願いします (kotoshi mo yoroshiku onegaishimasu), translated in this context as "Please support Morning Musume this year!" (Translation by HPS.)
Leaving Morning Musume before you get the wrong impression, here is a scene from Utada Hikaru's "UH Live Streaming". In this context it's her staff she is speaking to, so presumably she is already familiar with them. I guess this would be translated as a request for co-operation on todays task, translated as "Good luck today." (I don't have the info on who translated it at hand, will update later. The DVD linked to earlier is not subtitled.)
The last one is from the drama "Long Vacation". In this scene the parent (left) is asking the piano instructor Sena (right) to teach her daughter (even though it looks like a boy), or "take her under his wing" so to say. In this context it's translated as "please treat her well." (Translation by Japan-TV Fansubs.)
As you can see from the above examples, it's a very context-sensitive phrase when you step out of the "meeting-for-the-first-time" instances. If you have access to subtitled material, it's good to see how others have translated the phrase and see which best suit the context you are trying to translate.