What is the significance of the title of the play The Browning Version?
The "Browning version" is the translation made by Robert Browning of the Agamemnon of Aeschylus, which the teacher Mr. Crocker-Harris is having his students read in the original Greek. One of the students gives a copy of Browning's version to Mr. Crocker-Harris as a gift upon the latter's leaving the school.
The play (and film) has a parallel plot to the Agamemnon. Mr. Crocker-Harris's wife, like Agamemnon's wife, Clytemnestra, is having an affair with another man. Agamemnon, of course, is the Greek leader who engineered the victory over the Trojans. But when he returns home, his place has essentially been taken by his wife's lover, Aegisthus. Clytemnestra then murders Agamemnon. In a figurative sense, Mr. Crocker-Harris's wife's betrayal is a kind of re-enactment of this murder in modern terms. But the parallel goes much further. Mr. Crocker-Harris is a man whose time is past. He is an old-fashioned, inflexible teacher who bullies the kids in class and delights in showing them up and humiliating them. Like Agamemnon, his glory days, such as they were, are over.
If Terence Rattigan's play parallels the Agamemnon, why is it that the Browning version is specifically referenced? Probably because Browning's translation, while interesting and poetic on its own terms, is not considered highly by most classics specialists. It's old-fashioned and imperfect—like Mr. Crocker-Harris himself. The man has been disliked intensely by his students and the staff, but he is a wistful symbol of a previous age, just as Robert Browning's Victorian poems, as well as his Aeschylus translation, are.