The "Old Boy" line appears in Act I of The Crucible, and the line is spoken by Reverend Hale.
"We shall need hard study, if it comes to tracking down the Old Boy."
The "Old Boy" that he speaks of is the Devil (Satan). There is a small suspicion of witchcraft in Salem, and Parris requests Hale's presence to confirm that witchcraft is actually not present.
"There be no unnatural causes here. Tell him I have sent for Reverend Hale of Beverly, and Mister Hale will surely confirm that."
Hale has built himself a reputation for having quite a bit of experience rooting out "demonic arts." Hale shows up, and he has a lot of very heavy books with him. Parris comments on their weight, and Hale states that they should be heavy because they are weighted with God's authority. Parris then comments again about the books, and he says that Hale has come quite well prepared. Hale confirms this notion by saying that he and others will have to study hard to track down the "Old Boy" (Satan) and root him out of Salem. Considering the very real fear the Puritans have of Satan and demonic forces, it is quite surprising that Hale refers to the Devil in such a familiar and nonchalant manner. Between that comment and his comments about his books, Hale is presented to audiences as overconfident and a bit pompous.