Spoilers ahead!
IMHO, this is one of the five best Star Trek episodes of all time. Very few others ever truly capture Gene Roddenberry's original vision the way that this episode does. It is about finding a way to look beyond our way of thinking in order to see another's perspective. In this case, Captain Jean-Luc Picard must put himself in the shoes of Dathon, a Tamarian starship captain. The Tamarians are a not-unfriendly but careful and cautious alien race, as they force the Enterprise and themselves to understand each other using only the Tamarian language, which the Federation's best xenolinguists had been trying to decipher for years. It turns out that the Tamarians communicate verbally solely using allegorical and metaphorical references to their own culture's history. Even the Enterprise computer, with its universal translator, was for the most part at a loss when attempting to decipher the Tamarian language; it is able to provide only snippets of lore from the Tamarians' sector of the galaxy, but it turns out to be enough as Lt. Cmdr. Data and Dr. Crusher work on figuring out the language on the Enterprise, replaying a video of the Tamarians speaking to each other earlier in this episode, and observing their body language and tone of voice to determine intent or consequence, while Picard gets a live lesson in the Tamarian language, and ends up learning what Dathon is saying by observing his actions and comparing them to what he says. This works just enough for Picard to finally realize the correct linguistic cues to successfully determine the Tamarians' now formerly bizarre mode of speech. He is then able to report (after escaping Elidrel) to Dathon's first officer, in Tamarian, of what occurred on Elidrel between himself and Dathon, who perished on the planet.
The incident becomes a part of the Tamarian linguistic lore that their speech is drawn from, and they refer to it as "Picard and Dathon at Elidrel"—basically a repeat of the type of situation described in the episode's catch-phase: "Darmok and Jalad at Tanagra".
I have rarely seen an episode so well written and so powerful in the end. If you want to see the best that Star Trek has to offer, this episode is a must.