This page contains information regarding Star Trek: Lower Decks, and thus may contain spoilers.
The Tamarians spoke entirely by allegory, referencing mytho-historical people and places from their culture. As a result, when the Federation first made contact with the Tamarians, although their universal translators could successfully translate the individual words and sentence structure of Tamarian speech, they were unable to convey the symbolic meaning they represented. Without prior knowledge of the Tamarians' history and legends, a word-by-word translation was of no use to someone attempting to communicate with them. This language barrier led to the isolation of the Tamarian people after all attempts at communication had failed.
For example, instead of asking for cooperation, they would use a phrase such as "Darmok and Jalad at Tanagra", because their culture's stories include a tale of two Tamarians, Darmok and Jalad, who were brought together while fighting a common foe on an island called Tanagra. The problem with communicating in this fashion is that without understanding the context of the reference, the metaphor has no meaning. While explaining the structure of the language, Counselor Deanna Troi gave the example that "Juliet on her balcony" could be used to describe a romantic situation, but it is impossible to understand if the listener does not know who Juliet is, or why she was on the balcony. (TNG: "Darmok")
While he was trapped with Captain Dathon on El-Adrel IV, Captain Jean-Luc Picard gained an understanding of the metaphors used by Dathon as communication. Captain Picard used the Tamarian metaphors to establish communication with the Children of Tama and resolve a confrontation between the USS Enterprise-D and a Tamarian deep space cruiser. (TNG: "Darmok")
By 2381, the language barrier had been sufficiently broken for one Tamarian, Kayshon, to serve in Starfleet. The universal translator was largely able to translate Tamarian speech, though some metaphors were still translated literally. (LD: "Kayshon, His Eyes Open")
Some examples of the Tamarian language:
- "Arnock at the race of Natara" – running
- "Bazminti when he pulled back the veil" - an undercover operation
- "The beast at Tanagra" – a problem to be overcome
- "Children of Tama" – Tamarian
- "Chenza at court, the court of silence" – not listening
- "Darmok and Jalad at Tanagra" – cooperation
- "Darmok and Jalad on the ocean" – new friendship and understanding gained through a shared challenge
- "Darmok on the ocean" – Loneliness, isolation
- "Picard and Dathon at El-Adrel" – successful first contact between two alien cultures, or to work toward a common goal
- "Kadir beneath Mo Moteh" – failure or inability to communicate or understand; derisive in connotation
- "Kailash, when it rises" – a necessary loss or sacrifice
- "Karno in the forest with Mira" – overeating or weight gain
- "Kiazi's children, their faces wet" – downplaying the severity of a perceived injury
- "Kimarnt, her head cloudy?" – offering an intoxicating beverage
- "Kira at Bashi" – to tell a story
- "Kiteo, his eyes closed" – refusal to understand
- "Mirab, with sails unfurled" – signifying departure/engines to full/fleeing
- "Rai and Jiri at Lungha. Rai of Lowani. Lowani under two moons. Jiri of Ubaya. Ubaya of crossroads, at Lungha. Lungha, her sky gray" – greeting between two different cultures/races
- "Rapunki, when he joined the Seven" – greeting, expressing honor at joining a new group
- "The river Temarc in winter" – to cease an action, especially speaking; often used as an imperative
- "Shaka, when the walls fell" – failure
- "Sokath, his eyes uncovered/opened" – understanding/realization
- "Temba, at rest" – declining a gift. A gracious response signifying that the gift is unnecessary, or should rightfully be kept by the other person.
- "Temba, his arms wide/open" – signifying a gift
- "Unzak and Vhila as children?" – meaning unknown
- "Uzani, his army with fists closed" – to close rank and attack after luring the enemy
- "Uzani, his army with fists open" – to lure the enemy towards you by spreading your forces
- "Zima at Anzo" "Zima and Bakor" – danger/hostility arising from miscommunication/misunderstanding.
- "Zinda, his face black, his eyes red" – anger or conflict, also can indicate pain, possible indication of inability to survive (either self, or other party)
- "Zinda, his eyes red" – expressing pain or dismay
These phrases and idioms were often attenuated in conversation: "Shaka, when the walls fell" was heard shortened to "Shaka"; others followed a similar pattern.
Dathon also used "Callimas at Bahar" after experiencing pain in his shoulder, signaling to Picard with a hand wave associated with "stay back", or perhaps meaning "I feel better now" or "the pain is gone".
In devising the Tamarian language, "Darmok" writer Joe Menosky was inspired by three sources: the work of psychologist James Hillman (who had emphasized "All is metaphor"), the quote "Every word is a poem" from translator and poet John Ciardi, and the dense historical metaphors present in Chinese poetry and philosophical works such as the I Ching. (Star Trek: The Next Generation 365, p. 220)
In one scene where Picard attempts to treat a wounded Dathon, the Tamarian says "Kiazi's children, their faces wet". It is unclear what Dathon means by this, although (since Dathon is trying to shoo Picard away from caring for his injuries at the time) it may allude to children crying for no reason; the Tamarian may be saying that Picard should not worry or feel sad, as nothing could be done. It might also mean that Dathon knew he was dying, as Kiazi's children apparently knew, since Picard was trying to find out the extent of Dathon's injuries at the time.
Sfdebris states in his video "The Language of Darmok" that "metaphor" is likely not what is really going on with the Tamarian language by using Chinese as an example. He explains that in Chinese, "China" is two characters (one meaning "Jade" the other "Walls Surrounding") and the universal translator does a word-by-word translation. So instead of "China" it would produce the somewhat nonsensical "Jade surrounded by walls" or "Walls surrounding Jade". 
The Atlantic's "Shaka, When the Walls Fell" comes to a similar conclusion stating "Metaphor and image are not accurate descriptions of the Tamarian language’s logic." and presents allegory as a better representation of what is really going on which fits reasonably well into Sfdebris' explanation. 
The Tamarian language is explored further in the short story "Friends with the Sparrows" from the TNG anthology book The Sky's the Limit. In the story, it is explained that Tamarians have a fundamentally different brain structure to most humanoids, and as such experience concepts such as time and self differently.
The story also explains that Tamarian children learn the stories behind the metaphors, and thus their meanings, through enactment and repetition. Variations of meaning in metaphors were conveyed through subtle vocal and gestural cues that the universal translator had previously missed. In fields such as engineering and programming, a musical language was used to convey precise equations, numbers and instructions; thus explaining how Tamarians could effectively operate starships.