Genius Loci: Probably many, but Caradhras is perhaps the most active.
Genre Savvy: A lengthy conversation about characters in adventures occurs between Frodo and Sam on the steps of Cirith Ungol. They accept that they are at one of the darkest places in their story, but characters in a similar position in the old tales have overcome such difficulties, so there is still hope.
Sam: "That's not the way of it with the tales that really mattered, or the ones that stay in the mind. Folk seem to have been just landed in them... but I expect they had lots of chances, like us, of turning back, only they didn't. And if they had, we shouldn't know, because they'd have been forgotten."
Gentle Giant: Treebeard to Merry and Pippin, once he determines they are not orcs.
Getting Crap Past the Radar: A possibly unintentional example; while discussing the drowning death of Frodo's parents, out on the river by moonlight, the Gaffer seemingly implies they might have been doing something that caused their boat to overturn:
Gaffer: "Boats are quite tricky enough for those that sit still without looking further for the cause of trouble."
Go Back to the Source: Frodo returning the One Ring to Mordor, its place of creation, in order to destroy it,
God in Human Form: The group of so-called "wizards", while appearing as old human men, are actually five Maiar, a kind of angelic spirit, who are themselves incorporeal but can usually clothe themselves in any form they like. The five have been sent on a mission to help the peoples of Middle-Earth against Sauron, during which they are bound in their physical form, unable to change it, and also limited in their powers and knowledge.
Going to See the Elephant: Sam. He gets to see one, and realizes the Hobbit rhyme is an appallingly realistic depiction of a Múmak.
Good Hurts Evil: Shout A Elbereth, Gilthoniel at a Nazgûl and watch them writhe in agony.
Good Is Not Dumb: Frodo makes it very clear to Gollum that he is well aware that Gollum is trying to think of a way to betray the hobbits and take back the ring. Sam is surprised at this because he had assumed Frodo was far too good a person to be able to understand deception and treachery. Sam admits that he (and probably Gollum as well) "had confused kindness with blindness."
Denethor. He's certainly fighting to save Gondor from Sauron, but he's also a racist against "lesser Men" who treats Gandalf, Pippin, and especially Faramir like crud. What he was like beforeSauron messed up his head via Palantír, we never see.
Also, Frodo threatening Gollum with death in the Fire if he ever betrays the Ringbearer.
Gandalf, who has a short temper and a caustic sense of humor.
Half-Human Hybrid: Elrond and his family are half-elven, Aragorn's ancestors had elven blood (his ancestor was Elrond's brother), and the Uruk-hai are rumoured to be part-human part-orc.
Actually, if you trace the bloodlines of the Númenóreans back to Lúthien and then look at HER parents, you find that her father was an Elf Lord and her Mother was a Maia. As such, Aragorn, Elrond and company are part spirit/angel as well as elf.
Handicapped Badass: The Nazgûl have miserable eyesight during daylight and are blinded in bright sun.
Gríma Wormtongue is a student of Saruman's, and earlier uses similar Hannibal Lecture techniques on Théoden to render him helpless and hopeless against Saruman, and on Éowyn in order to break her resolve and drive her to desperation.
Have a Gay Old Time: Most obvious is the abundant use of 'queer', which is rarely used to describe something strange any more. Likewise 'gay' is used on a couple occasions to mean 'happy'. Some material also falls into Separated by a Common Language, since in parts of the world the term 'faggots' isn't any longer used to represent bundles of wood. Also, though it doesn't come up in the book, Celeborn's alternate name is the hilarious Teleporno (in Tolkien's planned but unpublished retcon of Celeborn's origin, he was not a Sindar Elf but a Teleri Elf; Celeborn is Sindarin for "silver tree", Teleporno is Telerin for the same).
The dwarf Groin, father of Oin and Gloin, though only if you pronounce it incorrectly. The orc Shagrat.
The song that Sam sings near the stone trolls, about Tom and the Troll, includes a nonsense word Tolkien made up: "boner". "Boner" at that time was a word meaning "error", although in said poem it seems to be just random rhyme-filler.
There are various subtler uses as well. Aside from writing seventy years ago, Tolkien was deliberately using a slightly archaic form of the language (at least when Men and Elves, rather than Hobbits, are talking.) For instance, when Aragorn says of the Nazgûl "they are terrible!", he doesn't mean they are of awful quality: he means they bring terror.
Healing Herb: Athelas is used by Aragorn, once in a failed attempt to cure Frodo of his Morgul wound and later successfully on those affected by the Black Breath of the Witch-King.
Helmets Are Hardly Heroic: Aragorn shows up to the Battle of the Pelennor Fields wearing a diadem (a royal heirloom of Arnor) instead of a helmet. Averted in an early draft, though, as originally he wore a proper helmet with a crown attached.
Here There Were Dragons: To us the War of the Ring takes place in such a world. To the characters, the earlier Ages were this.
Heroic Lineage: Elrond and Aragorn are descended from many of the Elven and Human heroes/protagonists from earlier Ages.
To lesser extents, Legolas, Gimli, Boromir, and even Pippin.
Hero of Another Story: There are hints of adventures that the other members of the fellowship had before meeting the hobbits at Rivendell, such as Aragorn's capture of Gollum, or Gandalf's escape from the Ringwraiths. At one point Sam wonders if Gollum thinks he's the hero of his own story.
The Appendices mention entire other battles - notably one at the Lonely Mountain - that occur completely offstage in parallel with the events of the novels.
Though interestingly enough, the Shire rather than any elf haven is the closest to this example. Along with being generally unknown to most of the world, they also generally don't bother themselves whatsoever with outside business.
High Fantasy: One of the most notable examples of the genre.
Hiss Before Fleeing: Gríma does this, along with spitting before King Théoden's feet, before returning to Isengard.
Honour Before Reason: The Three Hunters, who are determined to reach Merry and Pippin after their capture even if it is simply to sit down and starve with them. The various people who spared Gollum's life also apply, since letting him loose was much more dangerous than outright killing him, but in the end this illogical act pays off, since Gollum is crucial in the saving of Middle-Earth.
Hope Is Scary: Éomer's reaction to learning that Éowyn was still alive.
Also, Gandalf, who beat out Merlin in a "most iconic characters" poll in Britain once. And that was before the films...
Ignorant of the Call: Sam Gamgee refuses to accept the idea that he's any kind of hero. He sees himself as "just a gardener", even while he's in the middle of slaughtering orcs and beating Shelob into paste on his way to rescue Frodo.
I'm A Humanitarian: In the films, orcs will kill and eat one of their own if there's no other food. Averted in the novels where they express disgust at the thought of eating orc-flesh, but they rather enjoy eating humans.
They're Just Hiding is Treebeard's hopeful attitude, which is possibly backed up by a hobbit mentioning a "walking tree" in the north of the Shire.
Immortality Inducer: This is one of the powers of the One Ring, as experienced by Gollum. It is also implied that any of the greater Rings of Power would have this effect on mortals, such as the nine held by the Nazgûl.
In Mysterious Ways: According to Tolkien's letters, the trilogy is partially about Ilúvatar's guidance. It's only explicitly mentioned a few times, and none of God's names are ever even spoken; nevertheless, one could make a long list of the many, many suspiciously fortunate "coincidences" in the story — some very small, but ALL of which lead toward Sauron's downfall and the victory of the West.
I Shall Taunt You: Aragorn does this to Sauron by using the Palantír of Orthanc to show him Andúril (the sword that cut the ring from Sauron's hand), hoping to provoke Sauron into mobilizing his forces before they're fully ready. It works.