A friend recently asked me how I thought the major changes Peter Jackson made when adapting Lord of the Rings to the screen were received by fans of the book. Even with a few years perspective, I can only speak for myself and the fans that I know personally in saying they were well received. However, they were all also well-justified from a movie crafting perspective, so that definitely helped with the way they were perceived.
In general, I believe each of the changes was greeted with grumbling in some places, but the whole of the work was so obviously respectful of the original, and such an excellent movie for fans of the genre, that all the changes were forgiven. I'm also ignoring things that were cut from the theatrical releases, but included in the extended DVD versions (e.g. Galadriel's gifts).
Be warned, there are plenty of spoilers for both the books and the movies ahead, so if you're at all sensitive about such things, stop reading now :)
(And yes, I was able to name all these discrepancies off the top of my head... I probably missed some, though, since I forced myself not to look anything up on the web while writing it). Details after the jump...
Show a bit of urgency, people
In the book, 17 years pass between Bilbo's birthday party (Bilbo 111, Frodo 33) and the time when Gandalf tells Frodo to get out of the shire. Such a lack of urgency really wouldn't have worked for the movie, and the loss of the One Ring adequately explains Bilbo's rapid aging in the time he was at Rivendell before Frodo arrives. Most people I know thought the original time frame in the book was kind of dumb, so speeding it up for the movie met with general approval.
Removal of Old Man Willow, Tom Bombadil and the Barrows
Some people love this part of the books, many (*ahem*) see it as a relatively pointless detour that needlessly delays the arrival at Bree and the introduction of Aragorn as a character. You can get away with that kind of slow introduction in a long novel, but there was no way it was going to fit into the movie version of Fellowship. There was still plenty of tension in the journeys to Bree and Rivendell with just the Dark Riders as the threat.
Arwen stealing Glorfindel's lines
In the books, Arwen is actually portrayed as a princess type that actually doesn't do a great deal. Giving her Glorfindel's part in addition to her own gave the character more screen time and set her up as a far more suitable match for Aragorn. If they hadn't done this, I think people that weren't familiar with the books would have been (justifiably) up in arms about Aragorn choosing her over Eowyn. (Heck, even in the books Aragorn's choice isn't well justified - while Arwen is established as a generally competent lass in the Tolkien's notes, she really isn't portrayed as such in the books).
Also, it meant one less character to introduce within the time constraints of a movie (particularly since Glorfindel basically disappears from the story after Rivendell).
Aren't those separate stories?
The Two Towers actually has a fairly questionable story telling structure. The first half is the tale of what happened to Merry, Pippin, Aragorn, Legolas and Gimli, while the second is the tale of Frodo and Sam. Interleaving them was an obvious change to make for movie adaptation, and I don't think there was even any significant grumbling about that one.
Hey, let's throw Aragorn off a cliff!
Given the number of things from the books that were cut, this random addition was actually the change I found most irritating (and I'm not alone). In the books, nothing much of significance happens on the journey to Helm's Deep that I can recall.
That said, this sequence did give them a chance to practice a bit of "show, don't tell" as far as Eowyn's feelings for Aragorn went, so it made a certain amount of sense from that point of view. On the commentaries, I believe Jackson said they were concerned about the lack of threat to Aragorn for most of the movie and put this in to spice things up a bit.
Elves in Rangers' clothing
In the book, there are no elves at Helm's Deep. Instead, a bunch of Aragorn's fellow Rangers (i.e. long-lived Men of Numenor, like Aragorn) turn up. However, using the Rangers in the movie would have required setting up the Rangers as a group that existed, what their relationship was with Aragorn, and why their assistance was to be valued so much more than a similarly sized group of ordinary men. By using elves instead, Jackson was able to achieve a similar effect without trying to cram in all that additional exposition to established just who these people were.
Most people I know were happy to accept the troop of elves as the spiritual representatives of the Rangers from the books, but there was definitely some online griping about this one.
The outcome of the Entmoot
In the books, the result of the Entmoot is a decision to go destroy Isengard. After-the-fact persuasion by Merry, a curious perceptive blindness by Treebeard as to the state of the forests near Isengard and a subsequent unilateral decision to attack on seeing them in person are nowhere to be found. At this stage of the books, Merry and Pippin are still really just along for the ride - their characters arcs really don't start properly until Return of the King, and culminate in the Scouring of the Shire.
So, while I don't particularly like what Jackson, Walsh and Boyens did here (mainly because I think it disagrees with Tolkien's lore on what Ents can perceive within their own forests), I can understand the desire to give Merry and Pippin more of an active role earlier in the story, since the end of their pre-established arcs was going to be left out (more on that later).
Hobbits in Osgiliath
Another controversial one - in the books, Frodo and Sam are never in Osgiliath, and Faramir doesn't get there until some time in Return of the King. Instead, Faramir encounters the two hobbits and takes them back to his base in Ithilien. They there convince him of their purpose and he releases them to continue on their journey.
I suspect the screenwriters had two motives in making Faramir a bit more hardnosed about things and relocating the Ithilien part of the story. Firstly, it allowed the two halves of the story to be better weaved together at the end of second movie. As noted above, the book treats the two storylines as completely separate, which didn't really fit with the interleaved structure of the movie.
Secondly, it saved them the hassle of creating the Ithilien base as a location. (at least, I don't recall the movie making much, if any, use of that base).
I actually didn't mind this change, but a lot of people didn't like what they felt it implied about Faramir's character (personally, I thought making him a bit more hard-nosed than the book version wasn't a bad thing).
Giant spiders can't read calendars
The encounter with Shelob is actually part of The Two Towers. The screenwriters postponed it to Return of the King in order to balance out the screen time in the final movie a little better. This change also fit with the change to make their release by Faramir the climax of this half of the storyline in the second movie.
This was a change that made a lot of sense and I don't believe there were many complaints about it. Without it, The Two Towers ends on a serious downer with Frodo trapped in the tower, and their storyline in Return of the King consists of escaping from the tower and an awful lot of trudging across the Plains of Gorgoroth.
He's dead, Jim
Saruman didn't die at Isengard in the books. Instead, they get the palantir because Wormtongue throws it at them, and then they leave him in the tower under the guard of the Ents. This is important, since he plays a vital role in the Scouring of the Shire.
When they dropped the Scouring from the movies, they decided to make sure people knew he was quite assuredly dead by not only dropping him from a great height, but also making sure he landed on some big spikes that not only went all the way through him, but also were conveniently mounted on a wheel that could put him underwater as well. Just in case you were thinking he might have survived falling off Orthanc. I was somewhat surprised nobody carved out his heart and ate it on screen just to further make the point "No, he's really, really dead. He's NOT coming back at the end of the movie."
Pippin the pyromaniac
I believe the scene with Pippin sneaking up to light the signal fire was a new addition for the movie (although I don't recall what happened in the book instead - while Gandalf certainly argued with Denethor a lot about calling for the Rohirrim, I seem to recall Denethor grudgingly authorising the signal). I didn't mind it, since it gave Pippin more of an active role in the story, a necessary change due to the final (and likely least popular) change, which we move on to now...
The (non-)Scouring of the Shire
Wow, this one upset a lot of people. People that haven't read the books probably don't realise just how much Jackson cut from the denouement following the destruction of the ring.
It was a genuine loss that this had to go, but it really did have to go. The Scouring is a great story - it was the real payoff for Pippin and Merry's character arcs as they become the first true hobbit warriors the world has ever seen and take back the Shire from the nefarious villain that has taken control of it (who turns out, *gasp*, to be a certain prisoner that has escaped from Orthanc and wants revenge on those darn hobbits that ruined his schemes).
The other part of the Scouring is that after Merry and Peppin do the actual scouring part, the start of the rebuilding is what truly closes off Sam's story arc, as he takes the seed he received as a gift from Galadriel and plants it as a replacement for the party tree that was destroyed in their absence.
However, the problem with the Scouring from a movie makers point of view is that it is also quite a long story that takes place after the climax of the main storyline (i.e. the destruction of the ring and Sauron). Return of the King had a good dozen or so endings as it was, there was absolutely no way for them to also try to cram the Scouring in there.
In addition, the Scouring is all about the indomitable spirit of the hobbits when given a little bit of hope in the form of the mature and battle hardened Merry and Pippin and the ever-resilient Sam, despite the physical destruction and dominant presence of "Sharkey". That kind of thing is quite hard to portray on screen - an-indomitable-spirit-despite-surrounding-destruction looks an awful lot like plain old destruction to the viewer's eye.
So, they took the simpler option of leaving the Scouring out of the story altogether, with a homage to it included in Fellowship when Frodo looks into Galadriel's mirror. As noted in the relevant sections above, this also forced them to bring the development of Merry and Pippin forward a bit, and give them more to do in other parts of the story. (The bit with Merry stabbing the Witch King in the leg so Eowyn could kill him, though? Yeah, that was from the book - you know, with the whole prophecy thing and all...)
As I said above, I actually agree with their decision to drop the Scouring of the Shire. As cool a story as it is, it just didn't fit into the limitations of the movie format. Better to leave it out completely than try to jam it in where it didn't fit and do a poor job of it as result.