Draconic
Draconic is the language of dragons. Due to the longevity of dragons there is little dialectical variety and most dragons and dragonkin can speak to each other and be understood.
Due in large part to the influence of Lady Nyx, draconic has become an incredibly common language in Lugal and surrounding kingdoms to the point it has become more commonly spoken by humans than dwarvish.
Vulgar Draconic
So-called "vulgar" draconic also known as Koboldese is one of the few dialects of draconic. While largely mutually intelligible with the primary dialect of draconic, it has a looser syntax structure, slightly simplified vocabulary, and greater use of reduplication of syllables primarily for the purpose of emphasis. It also tends to have far more loan words from whatever cultures that a particular band of kobolds lives near, albeit adapted to kobold phonetics. Gold and silver dragons tend to find the kobold vernacular whimsical or childlike, but from a purely linguistic perspective this is not a fair assessment as it can express a complexity of thought equal to draconic if in a sometimes more cumbersome way.
Mechanically, any character who is able to understand the draconic language can understand kobolds speaking their dialect, but may need to pass a linguistics check with a DC of 10 for some particular or unfamiliar terms.
Scholar's Draconic
Also known as Wyrmriddle, Scholar's Draconic is to the draconic language as Druidic is to Common: A complex set of wordplay, poetic references, and archaic words which can be used to purposefully conceal information or confuse those who do not know it. It is not technically a dialect, but it does require a character to have learned it to be able to understand and is mostly known and used by ancient dragons, who use it to communicate with each other secretly similar to how Thieves' Cant is used by rogues. The conditions for a player character understanding it are up to the DM's discretion.
Tiamat's Tongue
A combination of Draconic, Abyssal, and alchemaical terms used by some evil dragons to conceal messages from other dragons. This dialect can be understood if the player character understands both Draconic and Abyssal and has proficiency in History. Despite its name, Tiamat cannot speak it.