Languages are tad-little-bit complex.
Human (or non-human (?) for my fantasy and sci-fi people out there) communication tends to have that characteristic: it’s weird and complex. There’s history behind them, the merging of cultures, the gears of societal expectations making them work together. They evolve, and change, and differ for many reasons, like the impact of geography, or society, or culture, or economics, or religion.
So, if you’re writing a fantasy or science-fiction project, it makes sense why you’d want to explore those details in your world. This doesn’t mean you have to build a whole language from scratch, with its own grammar rules, pronunciations and syntax. That’s actually… quite advised against, for the whole “falling into the endless pit of wordbuilding” problem, but I digress.
(If you ARE more of a “no, I need the whole language” kind of person and you will proceed to ignore my previous suggestion, a site you can use if you’re looking to actually build a fantasy language is Vulgarlang.com  which, according to their own website is: “a constructed language (conlang) generator for fantasy fiction writers and role players.”) I do, however, digress again.
There are ways to incorporate the feeling of linguistic importance in your story without creating the whole deal from scratch. But… Where do you start?
STUDY OTHER LANGUAGES
Pretty obvious, but if you’re only an English speaker... go outside the English barrier.
If you speak other languages and write in English, consider bringing some of the rules of another language into the new language.
SYNTAX: For example, I’m a native Spanish speaker, and in Spanish, we make things have smaller intent by suffixes called diminutivos. You can turn any noun into a diminutive if you try hard enough. In English, you don’t have a set of standard suffixes for making things little. But, you do get awesome suffixes like:
-ling (duck, but little, is duckling).
-y (doll - dolly)
-let (book - booklet) 
So, in my novel, I have a made-up word for the long phrase “my little runt of the litter”: skevling. Because in this case, I make the rules and everything can be a diminutivo if I want it to be. That’s the beauty of being your world’s god.
But you can consider other language rules, and borrow rules from other languages. Look up random grammar/syntax rules to borrow from other languages (PLEASE, do your research well. Don’t just take things at face value. We don’t want another “..and they had 20 words for snow” situation.) 
PHONETICS: This is especially helpful for descriptors later on. Is your language more guttural? Sing-songy? You can even google the different places and manners of articulation, and go from there. For example, the use of pharyngeal consonants. A pharyngeal consonant  is a consonant that is articulated primarily in the pharynx.
Arabic and Danish are examples of languages that have consonantal sounds considered “voiced pharyngeal approximant or pharyngeal fricative.” 
STUDY LANGUAGE FAMILIES
Understand the nature of language, the way it naturally flows into a tree diagram, developing and branching out over time.  You can explore how the proto-language of your world turned into other variants… and that’s why these two characters from neighbouring kingdoms kinda-sort-of understand each other, but not really.
Language is a fluid thing, with many variables. I recommend a nerdy Wikipedia rabbit hole: the list of language families.  Go find the language that kind of reflects the vibe you’re going for (But for the love of everything good and orderly, be respectful and do your research) and understand how it evolved over time, and how it branched out into other languages.
INCORPORATING IT INTO YOUR STORY
I’ve always said that there’s no greater source of inspiration than history. So, in my own personal opinion, I think that if you want to create the feel of a believable and relatable language, you should look at other existing or deceased real languages. (But that’s just my opinion, you are free to go down the Klingon  route and create something completely new.) So, let’s see how to incorporate it into your story.
There’s more fluidity than just Language A and Language B
I made this little simplified version of different linguistic expressions. This is just a small compilation of face-value information, please consider further research if a concept caught your attention!
Language: A system of arbitrary vocal symbols by means of which a social group cooperates. 
Accent: A manner of pronunciation peculiar to a particular individual, location, or nation. It can depend on the locality in which its speakers reside (a regional or geographical accent), the socioeconomic status of its speakers, their ethnicity, their caste or social class (a social accent), or influence from their first language (a foreign accent). 
Dialects: A regional variety of language distinguished by features of vocabulary, grammar, and pronunciation from other regional varieties and constituting together with them a single language.
The Doric dialect of ancient Greek
A dialect of Chinese spoken in Hong Kong 
Sprachbund: Also known as a linguistic area is a group of languages that share areal features resulting from geographical proximity and language contact. The languages may be genetically unrelated or only distantly related, but the sprachbund characteristics might give a false appearance of relatedness. 
Example: the Balkan sprachbund. The languages in question may belong to various separate branches of Indo-European (such as Slavic, Hellenic, Romance, Albanian and Indo-Aryan) or even outside of Indo-European (such as Turkish). 
Creole: A stable natural language that develops from the simplifying and mixing of different languages into a new one within a fairly brief period of time. 
Mixed Languages: languages in which whole component parts are from distinct language families or branches. In some, the vocabulary from one language and the grammatical system (phonology, morphology, syntax) from another are combined. 
Sign Languages: Languages that use the visual-manual modality to convey meaning. Sign languages are not universal and they are not mutually intelligible with each other, although there are also striking similarities among sign languages. 
Pidgin: A grammatically simplified means of communication that develops between two or more groups that do not have a language in common: typically, its vocabulary and grammar are limited and often drawn from several languages. It is most commonly employed in situations such as trade. 
So, if you decide your world wants a bit of the linguistic, consider the flow of the language and how your geography and history might affect the way people communicate.
You can consider also cant , jargons , argot , vernacular,  slang , accent, colloquialisms  or liturgical languages . Of course, not all at the same time, you can choose to expand on a few. Again, be watchful of the endless worldbuilding pit.
Naming characters and places thoughtfully is the ideal way to reflect the impact of language in your world. Take the opportunity and be purposeful with your names, that can truly add to the whole ambience.
Be coherent. If you’re writing a fantasy centred in 18th century Spain… Have Spanish names. Or, at least Iberian names. Please. Someone asked George R.R. Martin what his process was on naming characters. He takes common names and gives them a twist.  So, Edward becomes Eddard, and Danielle becomes Daenerys. That’s why they feel relatable, because a Britishly-based world has Britishly inspired names. Yet, the changes add to that fantasy feel.
(Minor spoiler for The Way of Kings) Have certain rules or things in common. One of my favourite little worldbuilding fun facts comes from Brandon Sanderson’s Stormlight Archives series. He incorporates this rule where symmetry is holy, so the names of the faithful in his world tend to be symmetrical, but never a perfect one, like Shallan (Because of sacredness, y’know?) However, the most impactful thing of this rule is that he doesn’t jump the first time to tell the reader this fun fact. He waits until it is useful, and serves the plot a purpose. Remember that your worldbuilding serves your plot, not the other way around, and linguistic worldbuilding absolutely fits into that.
Conclusion, or, as the kids call it TLDR: Language is complicated, and you can Be creative, there’s a lot of linguistic concepts that you can play around with. And… do your research! if you don’t, I will [insert threat here].
Thank you so much, Madison, for this opportunity! I hope this is helpful.
Follow Dani on Instagram: @writinggibsongirl
 Vulgar: fantasy language generator. https://www.vulgarlang.com/
 El Diminutivo en Inglés. https://www.inglessencillo.com/diminutivo
 The Eskimo words for snow cliché. https://en.wikipedia.org/wiki/Eskimo_words_for_snow
 Pharyngeal consonant. https://en.wikipedia.org/wiki/Pharyngeal_consonant
 Voiced pharyngeal fricative. https://en.wikipedia.org/wiki/Voiced_pharyngeal_fricative
 Language family. https://en.wikipedia.org/wiki/Language_family#Proto-languages
 List of language families. https://en.wikipedia.org/wiki/List_of_language_families
 Klingon. https://en.wikipedia.org/wiki/Klingon#Language
 Bloch, B. & L. Trager, G. (2020). Britannica. Language. Retrieved from: https://www.britannica.com/topic/language
 Dialect. https://www.merriam-webster.com/dictionary/dialect
 Sprachbund. https://en.wikipedia.org/wiki/Sprachbund
 Balkan sprachbund. https://en.wikipedia.org/wiki/Balkan_sprachbund
 Creole language. https://en.wikipedia.org/wiki/Creole_language
 Bakker, P. (2013). Mixed Languages. Retrieved from: https://www.oxfordbibliographies.com/view/document/obo-9780199772810/obo-9780199772810-0030.xml
 Sign Languages. https://en.wikipedia.org/wiki/Sign_language
 Pidgin. https://en.wikipedia.org/wiki/Pidgin
 Jargon. https://en.wikipedia.org/wiki/Jargon
 Vernacular. https://en.wikipedia.org/wiki/Vernacular
 Slang. https://en.wikipedia.org/wiki/Slang
 Colloquialism. https://en.wikipedia.org/wiki/Colloquialism
 Sacred language. https://en.wikipedia.org/wiki/Sacred_language
 George R. R. Martin on how he comes up with his characters’ names. https://www.youtube.com/watch?v=nTBQZBMQtOw