부탁하다: ask a favor of(take a good care of doing something, or so)
1. 잘 부탁드립니다. please take a good care of this. it can be any situation.
가) when you visit your daughter's teacher, you can say.
선생님 제 딸 좀 잘 부탁드립니다. Teacher, please take a good care of my daughter.
나) when you do contract with your client company to produce your products or so.
부장님, 이번 계약건 잘 좀 부탁드립니다.
Mr. Director, please take a good care of this contract btw us.
2. simply it means please.
김 과장님, 오늘 마감이니 서둘러 주십시오. 부탁합니다.
Section Chief(oftenly we call someone with his position with his name in korea) Kim, hurry up cuz today is a deadline. Please.
this '합쇼 form(-(ㅂ)입니다, -합니다 ending form) is extremely polite and formal, so we do use for business or very formal situation, such as TV news.
in casual way, 부탁해, can be same.
1. 내 동생 잘 부탁한다(when your sis is getting married to decent bloke, you can say to your future bro in law like this). please take a good care of my sis.
2. 빵 한 개만 사다줘. 부탁해. Buy me a loaf of bread. please.
so when you say '부탁하다' as a part of sentence, you can translate with take good care of something or someone. and when you use '부탁하다' seperately, you can switch with plz but it still means same. cuz 'plz' implies what you said before '부탁하다'.
hope it can helpful to you. have a great one. :)