Who is Sauron in the film.
The embodiment of evil in the world, seen clanking around commanding his forces in the Prologue, wearing a very spiky suit of armour. He made his presence felt via Saruman’s Palantír and his Servants.

Who was Sauron in the books
A lurking shadow, a sense of malevolent evil brooding in the tower of Barad-dûr in Mordor. Sauron is never seen in the books – his will is made known to the world through his Eye and his Servants.

Comparison for Fellowship Of the Ring
Prologue
The only sight we have of Sauron “in the flesh” is in the film’s Prologue, when his forces are battling the Last Alliance of Elves and Men. Sauron is first shown forging the One Ring in the fires of Mount Doom. This quick glimpse is followed by a scene of Sauron stepping onto the battlefield, carrying a mace, and with the One Ring on his finger. He is wearing a rather spiky suit of dull grey metallic armour, totally covering his face and body – apart from eye-holes through which can be seen a void.

Sauron kills Elendil, shatters Narsil, and then reaches down towards Isildur with the hand carrying the Ring. Isildur manages to slice off the finger bearing the One Ring, and Sauron then “disintegrates” – his armour falls to the ground and his physical body crumbles and disperses.

Bag End
Once book-Gandalf realises that Frodo’s ring is indeed the One Ring, he tells Frodo of its history. This tale included the history of Sauron, and his burning desire for the Ring.

“But last night I told you of Sauron the Great, the Dark Lord. The rumours that you have heard are true: he has indeed arisen again and left his hold in Mirkwood and returned to his ancient fastness in the Dark Tower of Mordor. … The Enemy is becoming very strong. His plans are far from ripe, I think, but they are ripening. We shall be hard put to it, even if it were not for this dreadful chance. The Enemy still lacks one thing to give him strength, and knowledge to beat down all resistance, break the last defences, and cover all the lands in a second darkness. He lacks the One Ring.” (FotR, The Shadow of the Past)

In the film, Gandalf is the first to see a flash of the Eye of Sauron, when bending down to pick up the Ring in Bag End. He then explains to Frodo exactly what Bilbo has left him:

Frodo: “But he was destroyed. Sauron was destroyed”.
Ring: (whispers) Isildur…
Gandalf: “No, Frodo. The spirit of Sauron endured. His life force is bound to the Ring and the Ring survived. Sauron has returned. His orcs have multiplied. His fortress of Barad-dûr is rebuilt in the land of Mordor. Sauron needs only this Ring to cover all the lands of a second darkness. He is seeking it. Seeking it, all his thought is bent on it. The Ring yearns above all else to return to the hand of its master. They are one, the Ring and the Dark Lord. Frodo, he must never find it“.

While both book and film show Sauron’s defeat by Isildur, followed by his gradual return to power, it is only the book that tells of his strength at the time of the Lord of the Rings – revealing that Sauron is strong, but not yet undefeatable. It is also only the book that explains why Sauron wants the Ring, and why he would move heaven and earth to capture it.

Bree
In the Prancing Pony, Frodo accidentally slips the Ring onto his finger in both the film and the book. However, it is only in the film that Frodo hears the voice of Sauron while wearing the Ring:
“You cannot hide! I see you! There is no life in the void…only death…”
In the book, the incident is much more muted, though several people do notice that Frodo disappears, notably one shifty-looking man.

Also in the film, Strider mentions Sauron the Deceiver to the Hobbits when he is telling them about the Nazgûl.

Council of Elrond
Both the film and the book use the Council of Elrond scene as a way to impart background information about the plot and its main characters, and the name of Sauron is mentioned in both book and film by Elrond during his discourse on the history of Middle-earth.

In the film version of Rivendell, there is a wall painting of Sauron smiting Isildur, and the reluctance to mention the name of the Dark Lord that is often seen in the books is certainly not present in the film:

Boromir: “The shards of Narsil! The blade that cut the ring from Sauron’s hand!”

Aragorn: “You cannot wield it! None of us can. The One Ring answers to Sauron alone. It has no other master.”
Boromir: “And if we fail, what then?! What happens when Sauron takes back what is his?! “

Gandalf: “Do you not understand that while we bicker among ourselves, Sauron’s power grows?! None can escape it!” (all from Council of Elrond)

The Eye of Sauron again appears in the film, filling Frodo’s mind when Gimli strikes the Ring with his axe.

The Ring Goes South
When trying to cross Caradhras in the book, Aragorn points out that there were evil things in the world that were not controlled by Sauron, and Gimli says that Caradhras was called ‘cruel’ long before the rumours of Sauron became common. There were no mentions of Sauron in this part of the film.

The Mirror of Galadriel
In the film, Frodo sees Sauron’s Eye while looking into the Mirror of Galadriel, and Sauron himself speaks to the hobbit, using the Black Speech. In the book, Frodo also sees the Eye, and a fantastic description of the Dark Lord’s gaze is given:

“But suddenly the Mirror went altogether dark, as dark as if a hole had opened in the world of sight, and Frodo looked into emptiness. In the black abyss there appeared a single Eye that slowly grew, until it filled nearly all the Mirror. So terrible was it that Frodo stood rooted, unable to cry out or to withdraw his gaze. The Eye was rimmed with fire, but was itself glazed, yellow as a cat’s, watchful and intent, and the black slit of its pupil opened on a pit, a window into nothing.
Then the Eye began to rove, searching this way and that ….” (FotR, The Mirror of Galadriel)

The book also explains that Galadriel has some sort of link with the Dark Lord, one small advantage for the Elves and Men – “I perceive the Dark Lord and know his mind, or all of his mind that concerns the Elves. And he gropes ever to see me and my thought. But still the door is closed!” (FotR, The Mirror of Galadriel) The true power of Galadriel is not brought across in the film. Yes, she is shown as a high-born Elf lady and the Sorceress of the Golden Wood – but is she shown as possibly the most powerful Elf in Middle-earth, and a crucial part of the fight against Sauron? No.

The Breaking of the Fellowship
Sauron is mentioned by Boromir in this film scene, when he accuses Frodo of taking the Ring to Sauron and betraying the Fellowship. This book chapter contains Frodo seeing the Eye while sitting on the Seat of Seeing at Amon Hen – a notable absence from an Eye-obsessed film.

Sauron and Saruman
New in the film is a regular interaction between Sauron (represented by his flaming Eye) and Saruman. Their conversations through Saruman’s palantír were short, sweet and to the point. For example:

Saruman: “The power of Isengard is at your command, Sauron, Lord of the Earth.”
Sauron: “Build me an army worthy of Mordor!”

Conclusions
The character of Sauron is very similar in the film and the book. He is the Dark Lord, skulking in Mordor, and sending only his servants out into the world to perform his dark wishes. What does differ is the portrayal of his character. In the book, and throughout Lord of the Rings, Sauron is not seen – he makes his presence felt through storm and clouds, black crows and shadowy Nazgûl. He works in the same way as some horror films – what you can’t see is scarier than what you can see. The feeling of menace overhanging Middle-earth is palpable.

The films, on the other hand, show Sauron more as a real person, rather than a shadow – it must be almost impossible to portray the subtleties of Tolkien’s intimations of evil on the film screen. Instead, Sauron’s menace is shown by his sheer size and hideousness, as well as the presence of his flaming Eye – wherever and whenever you least expect it.

Appendix: Comparison of the appearances of the Eye of Sauron

The Eye of Sauron makes frequent appearances in the film, and only a few in the books.

Film:
1. In Bag End – Gandalf sees the Eye
2. In the Prancing Pony, when Frodo puts on the Ring
3. At the Council of Elrond, Frodo sees the Eye when Gimli strikes the Ring with his axe
4. At the Mirror of Galadriel when Frodo was looking into it
5. When Frodo slips into the Wraith World after Boromir threatened him on Amon Hen
6. To Saruman – at Isengard

Book:
1. Lothlórien
2. Amon Hen (on the Seat)