Continuity mistake: When Pippin says "What about second breakfast?" he's crouched over by the pony, a bit away from Merry. Aragorn says a line and the camera goes back to the Hobbits. Now Pippin is further from the pony standing right by Merry, not crouched at all. He couldn't have moved that far during Aragorn's line and be standing so naturally.
Continuity mistake: After the hobbits take the tumble from farmer Maggot's field, they land in a heap with twigs in their hair. Frodo stands on the road and looks down it. He says, "I think we should get off the road" and the twigs are there. He looks back down the road and the smoke and leaves come swirling at him, he yells, "Get off the road" - no twigs. They hide from the wraith, then make their escape. Merry says, "What was that?" - and they have the twigs in their hair again.
Continuity mistake: When Frodo is pulling Sam out of the water at the very end of the movie, the underwater shots show Sam's hand grasping Frodo's wrist (Frodo's palm is against the inside of Sam's wrist), but as soon as he gets him above water, Frodo has a grip on Sam that forces Sam to keep his palm facing away from Frodo's arm (Frodo's palm is against the back of Sam's arm). Short of releasing Sam completely, there is no way to change grip like that in between the two shots.
Continuity mistake: As Gimli bows before Balin's tomb in Moria, the camera circles around the tomb and the body which Pippin later disturbs is visible in the background. It appears fairly substantial, is wearing a breastplate and has its head on its chest. However when Pippin touches it the body looks different; more skeletal and covered in cobwebs, with the head in a different position.
Continuity mistake: During the fight between Gandalf and Saruman at Orthanc, after Saruman slams backwards into the wall Saruman causes Gandalf to spin horizontally in mid-air. As Gandalf spins he loses grip of his staff, yet, in the next close-up as he hits the ground the staff is still in his hand.
Continuity mistake: When Legolas jumps up on a rock to look at the approaching flock of birds, you can distinctively see his breath showing. But at no other time do you see his, or anyone else's breath showing in the scene. (It occured to me that this might be smoke, but he doesn't have a pipe.)
Continuity mistake: When Gandalf sits near Frodo in Rivendell, he explains, "I was delayed." As he thinks back, in the first wide shot as Gandalf lies on the floor at the top of Orthanc, the bowl sits at the left side of the spire wall. Saruman forces Gandalf over the opposite edge and then onto his stomach towards the center of the floor. As Gandalf lies on his stomach and says, "And he does not share power!" in the next shot however, Gandalf is actually sitting up against the spire wall and the bowl is to his right.
Continuity mistake: During the prologue when Isildur and his men are riding through the woods, you see Isildur ride alone in front. Then the orcs attack, and when you see the orc jump at Isildur you can see a flag beside Isildur, meaning he is carrying it or someone beside him does. We previously saw him riding alone in front and when it cuts from the shot beneath Isildur we see the flag is gone and that there is no one beside him that is carrying the flag.
Continuity mistake: Not only do the arms on the two large statutes at Amon Hen change, the statutes themselves switch places. When we first see them, the statute of the younger man has a winged helmet and is on the left and the older with the smooth helmet is on the right. In the shot once the boats have gone past, the winged helmet statute is on the right and the smooth helmet statute is on the left.