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。