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). The Final Fantasy interpretation has four fingers on each hand.
See Special:Whatlinkshere/Etymology:Shinryu for a list of articles using this term.