The Return of the King
Sometimes we find it a little hard to figure Faramir out, because most of his character seems to come from what he isn't instead of what he is. He spends a lot of The Return of the King being defined against other people, including Boromir, Denethor, and Aragorn.
For Denethor, Faramir is not-Boromir. Boromir wanted war and enjoyed the glory of fighting, but Faramir is wise and likes book-learning. Faramir is still plenty tough on the battlefield, as we see in the war against Sauron. But he is still a lot less reckless than Boromir, and Denethor takes that as a sign of Faramir's lack of courage.
But if Denethor writes Faramir off for being less battle-hungry than Boromir, we also respect Faramir for his humility and his caution. He will fight when he has to, but he doesn't seek out battle the way Boromir did. In fact, that's exactly why Faramir survives the temptation of the Ring in The Two Towers and Boromir doesn't.
Obviously, Denethor totally rejects Faramir (at least, until Faramir is too injured to find out that he has finally won his father's love and approval). But the two men actually have a lot in common. Both are keen observers. Both can perceive people's motives. And both are wise and serious.
Gandalf even admits this similarity between Denethor and Faramir when he says: "by some chance the blood of Westernesse runs nearly true in [Denethor]; as it does in his other son, Faramir, and yet did not in Boromir whom he loved best" (5.1.92). Denethor and Faramir are both men of Westernesse: proud, stern, and smart.
But the major difference between Denethor and Faramir is also pretty clear: Denethor is so arrogant that he can't stand taking advice from Gandalf. And he cannot tolerate giving up his authority to the rightful king of Gondor, Aragorn.
Faramir, on the other hand, is wise enough to be less full of himself. Even though Faramir is his father's heir, he does not try to hang on to the rulership of Gondor, and he certainly doesn't toot his own horn. As the newest Steward, he happily agrees to step aside for Aragorn as the rightful king.
In fact, when Aragorn goes to the Houses of Healing to save Faramir from his terrible injury, Faramir's first words to him are words of loyalty: "My lord, you called me. I come. What does the king command?" (5.8.73). This response could not be more different from Denethor's decision to kill himself rather than give up Gondor to Aragorn.
It definitely says something about Faramir that Éowyn starts out with a huge crush on Aragorn, but then goes for Faramir when it becomes clear that she can't have the High King. We're not saying that Éowyn's love for Faramir isn't genuine. In fact, we think it's actually a lot more real than her crush on Aragorn, because it's based on actual interactions, rather than a pipe dream. But still, clearly these two guys have traits in common that make it possible for Éowyn to move on pretty smoothly from one to the other.
The thing is, Faramir is a lot like Aragorn but not quite as awesome. Pippin puts it really well when he first sees Faramir:
Here was one with an air of high nobility such as Aragorn at times revealed, less high perhaps, yet also less incalculable and remote: one of the Kings of Men born into a later time, but touched with the wisdom and sadness of the Elder Race. (5.4.39)
So, Faramir shares Aragorn's courage and "high nobility." But he isn't quite as elevated (or as "rarefied," to use a term that comes up in Frodo's "Character Analysis") as his new master. He's more human-seeming and approachable.
This like-Aragorn-but-not-as-great aspect of his character actually gives Faramir a huge advantage with Éowyn. It's around Faramir that Éowyn starts to relax a bit and lose her stiff-necked pride. As the two of them recover together in the Houses of Healing, Éowyn begins to laugh and enjoy her time with Faramir (which she never unwound enough to do with Aragorn).
Faramir may not be as glorious as Boromir, as proud as Denethor, or as kingly as Aragorn, but he makes his humbler personality work for him. After all, out of Faramir's immediate family, Faramir is the only one who (a) survives, and (b) goes on to rule something of his own (since once Aragorn is High King, he makes Faramir the Prince of Ithilien). Faramir's willingness to compromise gets him the girl and a principality to rule. That ain't too shabby for a dude who, a few chapters earlier, was almost burned alive.