JRR Tolkien is possibly the most influential fantasy writer of all time. However, he was also a product of his time, and from a modern perspective, there are tons of weird racist moments in The Lord of the Rings - both the novel and the movie series. That doesn't mean people shouldn't enjoy Tolkien's work - it has plenty of elements that are worth celebrating - but it's important to recognize the bigoted viewpoints that slipped into his beloved tales.
Racism in The Lord of the Rings isn't as simple as one character using a real-world racial slur to casually disparage an entire group. If the issues were that overt, everyone would have put the series down a long time ago. Perhaps unintentionally, Tolkien brought real-world prejudice into his work, and examining the instances of bigotry in The Lord of the Rings and its related works helps us recognize similar unconscious biases that exist in our society today.
Tolkien's deliberate association between dwarves and Jewish people is well-documented. He once noted, "[t]he dwarves of course are quite obviously, couldn't you say that in many ways they remind you of [Jewish people]?" This admittance makes his characterization of dwarves troubling - particularly in The Hobbit - as Tolkien relies heavily on Jewish stereotypes, which directly contribute to antisemitism.
While some dwarves are portrayed as good people, their preoccupation with gold and their incessant greed (which in part fuels the conflict in Lake-town) are inarguably offensive. To make matters worse, dwarves are portrayed as wicked in Tolkien's earlier work, which only serves to further harmful stereotypes about Jewish people that are still prevalent today.
Occasionally, characters in The Lord of the Rings break the light = good, dark = bad stereotype (Saruman comes to mind), but the prevailing trend is extremely troubling. The struggle of light versus dark is a concept that dates back to the early days of humanity, but in practice the model presents an uncomfortable binary that isn't useful for understanding contemporary conflicts (which are often morally gray all around).
Things get even worse when creators decide to demonstrate the concept by making evil characters' features appear darker. This is exactly what transpires in Tolkien's world; the heroes are beautiful and light-skinned, whereas orcs, Uruk-Hai, Easterlings, and Haradrim all feature considerably darker complexions. They're not evil because they're not white, but these associations tend to bleed into viewers' perceptions of real life, and can potentially cause them to be more critical of dark-skinned people.
While this isn't directly Tolkien's fault, perhaps if he'd included a few more characters of color who weren't evil in his works, it wouldn't have happened. British-Pakistani actor Naz Humphries told Stuff, an online publication in New Zealand, that she was turned away from an audition to be an extra in The Hobbit: An Unexpected Journey because she wasn't white enough.
"We are looking for light-skinned people," a film company representative said. "I'm not trying to be - whatever. It's just the brief. You've got to look like a hobbit."
According to Stuff, Robert Foster's The Complete Guide to Middle-Earth notes that the most common hobbits, Harfoots, are "browner" than other hobbits, but it seems the casting director (who was later fired) had a different, purely white vision for the Shire. Whiteness is equated with goodness so frequently in the franchise that perhaps it seemed consistent to call for only light-skinned actors, but doing so contradicted the source material.
In a letter from 1958, Tolkien wrote that orcs were "squat, broad, flat-nosed, sallow-skinned, with wide mouths and slant eyes: in fact, degraded and repulsive versions of the (to Europeans) least lovely Mongol-types." Even without this concrete evidence of Tolkien's bigotry, orcs' features are clearly racialized within the text, and the films do no better.
In addition to their black skin, many orcs also have dreadlocks. This combination of traits is clearly inspired by people of color, and the whole thing is utterly disgusting. Even if Tolkien's association was made subconsciously, these types of parallels condition audiences to associate blackness with evil, and help archaic stereotypes continue on into present day.