The Japanese language is the indigenous language of Japan, and a frequent source of both names and terms in the Teenage Mutant Ninja Turtles franchise. As the various versions of TMNT have largely taken place in the North America (particularly New York City), Japanese is not an everyday language or even a necessarily well-understood language to most of the characters, who instead speak English, which is also the language most untranslated versions of TMNT are produced in.
Most versions of Splinter as well as the Shredder can speak and understand Japanese as their native language. In some versions, the Turtles also have knowledge of Japanese, while in other versions they may know vocabulary related to ninjutsu but otherwise cannot speak or understand it conversationally. Even where they may have knowledge of Japanese, Splinter and the Turtles normally speak English together in their home.
For a sampling of some of this Japanese vocabulary, see the Japanese glossary.
In the Mirage continuity, proficiency in Japanese is not often addressed, but it is occasionally implied that all four Turtles are fluent in the language, though usually speaking English both at home and to others. This is retroactively cemented in the Usagi Yojimbo crossover story Namazu or the Big Fish Story, where it is specifically acknowledged that the four Turtles whom Miyamoto Usagi and Kakera had previously met had no trouble speaking or understanding Japanese, whereas none of the new Turtles they met in the story could speak or understand Japanese except for Michelangelo.
In the IDW continuity, the Hamato Sons were native speakers of Japanese in their past lives in feudal Japan. After they and their father Hamato Yoshi were reincarnated as turtles and a rat and then subsequently mutated, only Yoshi's reincarnation Splinter had full recall of his past life's memories, and the Turtles themselves could only speak English.
But in the mini-series Secret History of the Foot Clan, Michelangelo surprises both himself and his brothers by incidentally demonstrating his fluency in Japanese, as he flipped through the pages of the Ashi no Himitsu and could read and understand it effortlessly.
In a later mini-series, Turtles in Time, Mikey's brothers could still not speak or understand Japanese, and he had to interpret for them when they time traveled to feudal Japan and encountered the family of their own past lives.
This linguistic arrangement reappeared in the Usagi Yojimbo crossover story Namazu or the Big Fish Story, when initially only Mikey could communicate with Miyamoto Usagi and Kakera, until Kakera cast a spell that allowed everyone to understand each other's languages as if they were their own.
In embracing his Japanese fluency, Mikey became a collector of untranslated Japanese manga.
2012 TV series
In the 2012 TV series, all four Turtles are fluent in Japanese, but are almost never shown speaking it. In the season 1 episode Karai's Vendetta, April O'Neil mentions having learnt some conversational Japanese from "her brothers" which she speaks occasionally with Mr. Murakami. The Turtles' fluency is not fully implied until Tale of the Yokai, when they time travel to meet Tang Shen in the days before her death, and are covert witnesses to the Hamato Clan's fall to Oroku Saki and the Foot Clan. All scenes involved, however, are depicted in English.
Japanese dialogue is generally rare in the series, save mostly for the Japanese commands Splinter speaks to his children. But Splinter himself occasionally drops Japanese words, like "nani?!" when startled or "moshi moshi?" when answering the Cheese Phone. Among other characters not specifically (or immediately) linked to Splinter's backstory, Karai and Tiger Claw are also specifically shown to be fluent in Japanese, which is not surprising as both are natives of Japan.
The Amazing Adventures comics series, which is a separate continuity originally based on the 2012 TV series, the story departs with the TV series in establishing early on in the story arc Zodiac that the Turtles cannot immediately understand Splinter and the Shredder when they briefly speak to each other in Japanese, and are only understood with the help of translation software on Donatello's T-Phone.