SARUMAN: Concealed within his fortress, the Lord of Mordor sees all. His gaze pierces cloud, shadow, earth, and flesh. You know of what I speak, Gandalf: a great eye, lidless, wreathed in flame.
This eye that Saruman speaks of isn't really a big part of the books. The "searching for the Ring" that the book talks about Sauron doing is all a metaphor. Sauron wants the Ring and uses the Nazgûl to retrieve it—but it's not like he's sitting in Mordor with a magnifying glass, peering into the corners of Middle-earth looking for a tiny Ring-bearing hobbit.
But in the movie that's (almost) exactly what he appears to be doing. The eye is an invention (or expansion really) by Jackson to give Sauron and his searching a visual, visceral quality; something that viewers and characters alike can see and be frightened of.
And it's not just Sauron who's doing all the seeing in Fellowship. Saruman has a Palantír, also known as a Seeing Stone (but literally translated to "farsighted" or "one who sees from afar"). The Palantír can be used to connect with other Palantíri and thus used for various levels of long-range communication… or spying and deception and other evil things.
But sometimes sight is more metaphorical. For example, Galadriel can "see" all kinds of things. She doesn't literally see them, but she has knowledge of the future and of what evil lurks in the minds of men, hobbits, dwarves and elves. But to underline this knowledge, we get more than a few ominous close-ups of Galadriel's sparkling eyes.
Frodo also uses Galadriel's silver basin to see—this time literally—into a future Shire. This basin shows him "that which has not yet come to pass." But Frodo isn't just reliant on Galadriel's magical birdbath: he's also given visions of Sauron's eye whenever he puts on the ring.