Once a promising kung fu master trained and raised by Master Shifu, Tai Lung is a strong intgelliant, and skilled warrior. His only thing in life is to prove that he is the Dragon Warrior of legend.
Kung Fu Panda movie
Found as a orphan baby who was left at his doorstep, Master Shifu raised and trained the snow leopard as his son. The master believe Tai Lung to be the legendary "Dragon Warrior", even naming him Tai Lung (Great Dragon). However, when Master Oogway believed Tai Lung was not the warrior and didn't not let him have the legendary scroll to teach him the ways, and Shifu siding with him despite all the years of training and building Tai Lung up; Tai Lung, however, turned violent and went on a rampage until Oogway used acupuncture to stop him.
Tai Lung was then sealed in a special restraint deep in a mountain prison with a army of guards on him... only to escape using a goose feather (a goose sent by Shifu himself to ensure Tai Lung was still there) and his own speed and skills.
As Tai Lung made his way to the Jade Palace, he fought and won against the Furious 5. He then fought and won against his Master Shifu, having his anger and ego overlook his Master pleas of forgiveness for turning his back on Tai Lung when Oogway said he wasn't the Dragon Warrior. Tai Lung then claimed the scroll that will teach him to be truy be the Dragon Warrior... only to find it blank.
Enraged by this, Tai Lung went a destructive rampage through the town, only to confront with the actual Dragon Warrior picked by Oogway, Po the Panda. Tai Lung's tactics did not work on Po and the panda's crazy style constantly threw Tai Lung off, the snow leopard was soon loosing the battle until Po had him the Wuxi Finger Hold, which ultimately defeated him.
Although he only mention/cameo in the sequel movies, he reappears in the video games, but not much is know to what "officially" happened to him since the Wuxi FInger Hold.