The death of the white wizard Saruman differs slightly from Tolkien's original book Return of the King to Peter Jackson's film adaptation of the same name. But although the whereabouts and the circumstances change between the two, both versions agree that Saruman is killed by
In the aftermath of that battle Frodo confronted Saruman and exiled him from the Shire, but before he could leave, Gríma Wormtongue killed Saruman by slitting his throat with a dagger, on the very doorstep of Bag End.
Gríma killed Lotho Sackville-Baggins and may have eaten him, which he claimed to have done under Saruman's orders. Spurred by the words of Frodo that he did not have to follow Saruman, and having been pushed over the edge when Saruman scorned him, Gríma used a hidden knife to slit Saruman's throat, killing him.
As in the book, Gríma kills Saruman, but by stabbing him in the back, not slitting his throat. Saruman's body then falls from the tower and is impaled on a spiked wheel, one of the remnants of his infernal machines. Gríma himself is shot by an arrow fired by Legolas, thereby mirroring his death in the book.
Jackson's reasons for cutting Saruman's demise were twofold; firstly, to pare down a film that was already testing the limits of how long a person can comfortably sit on a theater seat, and secondly because of the narrative flow.
In the end, the diminished Saruman is murdered, his throat cut, and Shippey notes that when he dies his spirit "dissolved into nothing".
Isildur took up the hilt-shard of Narsil, Elendil's sword, and cut the One Ring from the hand of Sauron. Despite the urging of Elrond and Círdan, Gil-galad's lieutenants, Isildur did not throw the Ring into the fires of Mount Doom. He made a scroll with a description of the Ring and a copy of its fading inscription.
The fact that Grima's life is spared is consistent with Tolkien's theme of mercy and not dealing death in judgement that runs throughout his work. The choice to have Aragorn specifically save Wormtongue in the movie seems done for drama and to emphasize that Aragorn in particular is wise and merciful.
Tolkien describes him as "a wizened figure of a man, with a pale wise face, and heavy lidded eyes", and a "long pale tongue". Gríma was widely disliked in Edoras; everyone except Théoden called him "Wormtongue".
The reason why Grima cried was because he realised that Saruman intended to kill all of the people of Rohan including the woman he was in love with - Eowyn.
Gollum was a Stoor Hobbit of the River-folk who lived near the Gladden Fields. In The Lord of the Rings it is stated that he was originally known as Sméagol, corrupted by the One Ring, and later named Gollum after his habit of making "a horrible swallowing noise in his throat".
Gandalf sent the Mouth of Sauron away as they had not come to bargain with a servant of the Dark Lord.
Short answer: It was a reflex. The director decided Legolas wanted to murder someone because he was one short of Gimli's count.
It soon became clear that Saruman desired to possess the One Ring himself. He soon found that Sauron had more knowledge of the possible location of the One Ring than he expected, and in T.A. 2941 he finally consented to an Attack on Dol Guldur against Sauron at which point Sauron retreated to Mordor.
Whatever his motives were for the killing of Saruman, Grima pays for it with his life. In the film version, he is shot by an arrow from Legolas, too late, and in the books, he is shot by three arrows by a band of hobbit archers trying to protect The Shire.
No One Took Saruman's Power - He Squandered It
Even still, Gandalf gave him a chance to repent after the Battle of Helm's Deep, but the traitorous wizard refused. So, Gandalf expelled him from the order of the Istari.
The Two Towers
By this point Théoden had grown weak with age, and was largely controlled by his chief advisor Gríma Wormtongue, who was secretly in the employ of the corrupt wizard Saruman.
Gríma Wormtongue in The Lord of the Rings film series. Wormtongue was once a man of Rohan, but betrayed them for Saruman, and they were manipulating Théoden, the king of Rohan. It is strongly implied that he murdered Théoden's son Théodred by poisoning him in his sleep.
Stormcrow was a name given to Gandalf by King Théoden in Rohan, a reference to his arrival being associated with times of trouble.
Robin is an avatar character whose name, gender, and appearance can be customized by the player. In the original timeline, after Robin and Chrom defeat Validar at The Dragon's Table, Robin becomes possessed by Grima and kills Chrom.
The books imply stalking and watching. The films add controlling and manipulative words. Grima could never overcome a shieldmaiden, but he had free reign to stalk and stare, and it's possible he tried to wile her into submission the way he did with Theoden. He obsessed over her, and she rejected him.
Grima was originally a normal advisor to Theoden who was possibly a good person. He had genuine feelings to Eowyn and originally wanted what was best for Rohan. However, because of his appearance and seemingly weird nature, he was hated and disliked by majority of the people of Rohan, especially Eomer and Eowyn.
Since their will was not bound to Sauron now, it is more likely that they ran away in terror and scattered across Middle-earth. After Sauron's defeat, Aragorn and his army killed the remaining Orcs in Mordor.
Aragorn is not half Elf, although he is a descendant of Elros, who is half Elf (and the brother of Elrond, the half-Elf who raised him), which explains why Aragorn's life span is unusually long.
The white of the hand suggests a connection with the Wizard's traditional colour and title of 'Saruman the White' (though by the time of the War of the Ring, Saruman was referring to himself as 'Saruman of Many Colours', so even this connection is open to question).