Shinryu is an Original Creation monster bred at the Monster Arena in Final Fantasy X. It resembles the Guardian of Bevelle, Evrae. The fight is held underwater, and as such, only Tidus, Wakka and Rikku can participate. Shinryu can be fought after at least two of each of the underwater fiends in the Mt. Gagazet caves are captured.
Shinryu counters hostilities with a physical attack and often uses Eraser, which petrifies (unblockable even with a Stoneproof or Ribbon armor equipped) and shatters one victim instantly, removing them from the battle. Shinryu will stop using this attack if there is only one party member left. Another dangerous attack is Shining, which inflicts heavy non-elemental magic-based damage with 8 hits in random characters.
Shinryu drops weapons with Double AP and armor with Auto-Med. It also drops two Wings to Discovery, needed for customizing Break HP Limit on armor. Its equipment drops provide a hidden 6% critical hit chance bonus instead of the usual 3% (this is relevant to weapons and armor pieces, but only to attacks that deal physical damage).
Shinryu is one of two bosses considered impossible to defeat in No Sphere Grid challenge runs, even when Zanmato is allowed (it is not possible to use Zanmato on Shinryu). The only other boss deemed undefeatable in such runs is Penance. By extension, Nemesis, who can only be fought after defeating Shinryu, has not been defeated in a No Sphere Grid run.
It is best to fight Shinryu after the party has maxed out many of their stats (max Luck and Magic, or HP of 9999+ are not necessary). This way, Shinryu's counterattack and magic attack Shining will do less than 1000 damage even without Shell and Protect.
The party should start using Hastega, and hack away using Quick Hits, which will be countered. If Rikku, Tidus and Wakka have obtained their Celestial Weapons, and the party is dealing 99,999 each hit, the battle is not much trouble. After two characters have been erased Shinryu will not use Eraser on the last one, in which position it would be ideal to cast Auto-Life and keep using Quick Hit.
Shinryu is the Japanese on'yomi name for Shénlóng, literally "god/divine dragon", an important ancient dragon in Chinese mythology. It is said to preside over storms and rain, and can bring disastrous drought, thunderstorms and general bad weather if it feels neglected and/or angered.
A notable difference between common Chinese dragons and a Shenlong/Shinryu is their fingers; the latter have five on each hand as opposed to four, and they are said to share the same green-blue-ish color as another mythological dragon, the Qinglong/Seiryu (lit. Azure/Blue Dragon).
While the most common Final Fantasy interpretations have four fingers on each hand, other versions also possess five-fingered hands like with the original Shenlong/Shinryu, which mainly apply to the stronger ones.