This whole article, a small novel, two abridged textbooks and an epic poem could be written about the sheer volume of deleted scenes from the Lord of the Rings trilogy, but there's one particular cut from The Two Towers that seems particularly important. The flashback scene we're referring to shows Boromir and Faramir having a conversation after stomping some Orc ass--a flashback to before anybody knew that a certain hobbit had found a certain ring. Then their father, Denethor, shows up to kill the moment, because he's just generally not a "party type of dude."
Boromir (Left), Faramir (Middle), Denethor (Right). Because all Gondor boys look alike.
Boromir and his father Denethor discuss the Council at Rivendell, and the fact that "Isildur's Bane" is the One Ring (a fact not presented otherwise). Denethor instructs his son to retrieve the Ring for Gondor.
Which went over well, in case you're wondering.