毕竟 means “after all”, for example : He is your father after all. 他毕竟还是你的父亲。 究竟 has a common meaning of “what on earth” , for example What on earth is that! 那个究竟是什么？ Or there is another meaning as “to the bottom of sth”, 问个究竟，ask to the bottom of it. 到底 has more or less the same usages as 究竟 but in the second meaning the word 个 is always used in front of 究竟 but not with 到底。For example, 问个究竟 说个究竟 问到底 说到底。 终于 means “finally ” for example finally I am here 我终于到了。

They are not synonyms. Look at these example sentences: 1. 毕竟: after all 不要责怪他了，毕竟他只是个小孩。 Don't blame him, after all he's just a child. 2. 究竟: actually, exactly 他究竟发生了什么事？为什么他几天也没上班？ What actually happened to him? Why hasn't he come to work in days? 到底 is a colloquial synonyms of 究竟. It is only used in speech not written Chinese. 3. 终于: finally, eventually 经过五个月的调查，警察终于找到了小偷。 After five months of investigation, police finally found the thief.

我不认为用英语能真正解释的清这些词。