Parental Favoritism: Boromir (the elder son) is heavily preferred to Faramir by their father, Denethor. It's especially emphasized in The Movies, where Denethor is shown as blatantly unfair; in the book, Gandalf at least believes that it is partly that Denethor is still grief-stricken over the death of their mother.
And being Mind-Warped by the palantír.
Parental Marriage Veto: When Elrond finds out that Aragorn and Arwen are in love, he sets down what seems to be a impossible set of restrictions on their marriage (Sauron must be vanquished, Aragorn must unite the ancient kingdoms of Gondor and Arnor and become High King once again). Needless to say, Aragorn helps fulfill every single one of these conditions, and Love Conquers All.
Not really so much a Veto as just a traditional dowry: the daughter of an elven lord of Elrond's stature can only be married to a very important person, after all, and in the book Aragorn's final goal was to become king anyway, so it wasn't that tall an order. Who wouldn't want to provide the safest and best environment for their kid's future?
It also harkens back to The Silmarillion when Thingol tasked Beren with retrieving a Silmaril from Morgoth's crown before marrying Lúthien. Since Aragorn and Arwen are both descendants of Beren and Lúthien it seems appropriate; at least to Elrond.
Party Of Representatives: Invoked at the Council of Elrond which calls for the Fellowship of the Ring to be composed of representatives from all free peoples of Middle-Earth.
Passing the Torch: Bilbo to Frodo, although not without one last attempt by the elder Mr. Baggins to join the new adventure.
Past Victim Showcase: What the Mouth of Sauron hints at when he shows Frodo's mithril shirt, his elven cloak, and Sam's sword (which Sam had switched with him when he presumed the other hobbit dead) to the Lords of the West at the Black Gate.
Pet the Dog: Gollum has a rare moment of humanity when he catches Frodo and Sam asleep on the stairs of Cirith Ungol. He even attempts to pet Frodo's knee. Unfortunately, the moment doesn't last.
High Fantasy: It popularized the genre and is generally credited with creating it, although high fantasy in the novel format is actually older than Tolkien. The demand for novels similar to Lord of the Rings was so great that many imitators joined in to feed the demand. The term "Tolkienesque" has been used to describe the literature of his many imitators. A few writers actually tried to go in a different direction than Tolkien, such as Ursula K. Le Guin with her Earthsea novels. Even today, Tolkien's shadow is so big that it's difficult for a writer to escape it.
Hippies: The slang term "weed" sprang from a certain common misconception / bit of Fanon concerning Hobbit horticultural habits (Jossed by Tolkien himself).
Role Playing Games: The Fellowship can be seen as the prototypical RPG party. It established many archetypes and tropes that are seen RPGs like "rangers", warrior dwarves, the Balrog and Mithril. The Fellowship's trek through the dwarven city of Khazad-dûm might have been the basis for Ruins for Ruins' Sake. It inspired D&D and many of the RPGs that came after it.
War Gaming: Before LotR, war games were limited to historical wars like WWII, the Napoleonic Wars, and the American Civil War. LotR popularized the idea that war gaming can take place in a fictional land with fictional races and nations. It influenced such games as Warhammer, Warcraft, and StarCraft.
Which came full circle when Games Workshop made a tabletop wargame based on LOTR, inspired partly by Warhammer.
Possession Implies Mastery: Heavily, heavily subverted. About the only thing you with do with more possession of the ring is to become invisible. It may not even be possible for most ordinary beings to master.
Post-Climax Confrontation: The One Ring has been destroyed along with Sauron and the armies of Mordor, Aragorn has been crowned the king of Gondor, and the members of the Fellowship have separated to return to their homelands. When the hobbits return to The Shire, however, they discover it has been taken control of by Saruman, and they have to fight one last battle against him.
Post Victory Collapse: Merry and Éowyn after defeating the Witch-King, Sam after defeating Shelob, and Frodo and Sam when they finally achieve their goal (though they do manage to make it down the mountain first).
Merry and Pippin are kind of lampshaded Proud Warrior Race Guys . Tooks and Brandybucks have a reputation among other Hobbits for being more adventurous and warlike as Hobbit clans go, but Merry and Pippin just act like Hobbits. At least at first. Later both kind of go native and become something like Hobbit Swashbucklers. Yeah, I know, what would a Hobbit Swashbuckler be like?
Publisher Chosen Title: The publisher meddled with the titles of the three volumes. Tolkien wanted the last one to be named The War of the Ring to avoid spoilers, but it didn't get through.
The Quest: Frodo's quest is a double subversion. So much so, it might called an Anti Quest.
Rags to Royalty: Aragorn, sort of. He was already the leader of the remaining Dúnedain in the north, but they collectively appeared to be this to most other humans. Among the elves, however, he and his people still had fairly high status. And when his father died early in his life, he was taken in and raised by one of the most important elven leaders still in Middle-Earth.
Ransacked Room: Happens to Frodo's house in Crickhollow after he leaves.
Every last named elf is this as well, with Círdan being the oldest of all elves that haven't set sail to Valinor. (It's probably relevant that Círdan is the only Elf in the entire story who's explicitly stated to have grown a beard, let alone a long white beard.)
Reasonable Authority Figure: It takes Gandalf some ingenuity to restore Théoden's past judgement, but after Gandalf succeeds the King of Rohan is one of the main assets to the Free Peoples. Gandalf tries later to talk sense into Denethor, this time to no avail.
Recurring Dreams: Faramir, twice. One of these is based on the 'great wave' dream that both Tolkien and his son Michael had.
Reforged Blade: Narsil, sword of Elendil, which was shattered - but the hilt-shard cut The One Ring from the finger of Sauron, thus winning the war. Much later, they were reforged for Aragorn's use and renamed Andúril. He never does anything 'special' with the sword, but since it serves as a symbol of his status as the Returned King, and since the reforging was part of a series of prophesied events leading up to the final fall of Sauron, it qualifies as a Sword of Plot Advancement.
Refused by the Call: A 128-year-old Bilbo Baggins is the first to offer himself as Ringbearer for the Quest of Mount Doom, but is gently refused by Gandalf and Elrond as too old and too vulnerable to the Ring.
Though he actually has a good precedent for not accepting Aragorn's claim - especially since he has reason to believe Aragorn won't act in Gondor's best interest - and we don't know what he would have actually done had he not been Driven to Suicide. So it's certainly not a clear-cut example.
The Remnant: Saruman qualifies in a round-about sense, in that he is a former 'Evil Overlord' , but reduced to a pathetically small scale after his armies are routed and he is cast out from Isengard. He spends the remainder of the book running the Shire into the ground, turning into a sort of bandit leader with a mob of 'ruffians'. He is stabbed in the back (completely literally) by his servant at the end.
Rhyming With Itself: One of the parts of Sam's Oliphant poem rhymes "Me" with itself, but it's worth noting the poem is more whimsical than most others throughout the story.
The Rohirrim avenging Théoden's death on the Pelennor Fields.
Rousing Speech: Given by Aragorn at the Stone of Erech and Théoden before the Battle of the Pelennor Fields. Subverted in Éomer's speech during the same battle after he finds the bodies of Théoden and Éowyn, which is known to fans as the "Let's All Go And Get Killed" speech.
Royal Blood: Flows in Aragorn's veins and is rather important.
Royals Who Actually Do Something: Aragorn again, as well as Théoden, Théodred, Éomer and Éowyn. Legolas and Imrahil are princes, and Boromir and Faramir are sons of the steward, Denethor, who is in turn descended from Anárion, brother of Isildur. Merry and Pippin are eldest sons of the rulers of Buckland and the Shire, respectively.
The same goes for Brand (King of Dale) and Dáin II Ironfoot (King Under the Mountain), whose deeds are mentioned only in the appendices.