When it comes to symbolism, this is the One Symbol To Rule Them All. It's true, we don't see nearly as much of the most infamous finger accessory in the second movie, but when we do make it to Frodo and Sam's storyline, the presence of the Ring is always front and center.
The Ring is a symbol of greed. The Ring doesn't do nearly as much speaking as it did in the Fellowship, but when another man comes into contact with it, the Ring begins talking to him immediately. Men are known for their lust for power and fame, and Faramir, as true-hearted as he may be, is no different.
And to Faramir, the Ring is not only a way of destroying the enemy; it's also a way of regaining his father's trust: "a chance for Faramir, captain of Gondor, to show his quality." All Faramir needs to do is complete the task his brother could not—give Gondor the power of the enemy—and surely this'll make his Daddy dearest proud of him.
The Ring is also a symbol of obsession and dependence. We thought the Ring had warped Bilbo when we met him so long ago in Rivendell, but after seeing Gollum we realize just how tightly the Ring can wrap someone around it's metaphorical finger. Despite Frodo's best attempts to win back the original Sméagol through kindness and trust (and not getting him killed in a sacred pool), the Ring will always be Gollum's precious. It's consumed his body, distorting his hobbit like form and turning him into a crawling, bony, big eyed… thing. After half a millennium, Gollum doesn't even seem to care for himself—he only cares about his preciousssss.
Finally, the Ring seems to symbolize a third thing, although it's more elusive. Frodo always speaks of a heaviness that seems to be not only an emotional or mental weight, but a physical one as well. To him, the Ring is a burden—but what does this mean? Frodo doesn't seem to be driven by greed (and the Ring isn't his precious), but it does at times seem to possess him. Mostly this means it makes him irrationally angry, but when he has a sword at Sam's throat, we know things have gotten way out of control.
Whatever this heaviness is, we hope it never reveals itself until the Ring is sitting in the hot fires of Mount Doom.