What is the English equivalent of 拜托, and what is it's proper usage in Chinese?
The literal meaning is 'to respectfully ask someone to do something', and sometimes it's used in that way. But often, even though it's used as a verb, 拜托你xxx can just be translated as 'please can you do xxx for me'. It's not as common as 'please' in English though, so it sounds more formal / polite.

You can also use it semi-sarcastically to show your exasperation with someone - nciku.com gives 拜托你闭嘴 (please just shut up) and 拜托你少管闲事 (I'll thank you to mind your own business).

As well as being a verb, you can use it on its own or at the end of a sentence. As above, this is either a polite phrase like "I'm begging you for your assistance" or a sarcastic phrase meaning "oh for God's sake" depending on context.