Before Kung Fu Panda
While her origin is mostly unknown the episode Kung Fu Daycare from Kung Fu Panda Legends of Awesomeness shows that she was placed at the steps of Bao Gu Orphange by her birth parents.
She lived there in isolation because everyone feared her power and temper. One day she escaped her room and went to play with the other kids. Her claws and teeth scared them however and her temper again got the better of her. When put back in her room she agreed with everyone elses' sentiments that she was a monster.
With no clue what to do the orphange summoned Master Shifu.
Master Shifu was exactly what she needed and told her that despite what she and everyone thought, she wasn't a monster. He then taught her dominos and despite frustration from constantly breaking them she eventually learned discipline.
After training like this for a few months, she was able to place all the dominos on a table and showed them her control winning the friendship of the caretakers and fellow orphans.
Even after all that, people still didn't want to adopt her because they were still scared and she was left at the orphange.
Shifu one day arrived and said for her to come home and that was the day she went home to the Jade palace.
Kung Fu Panda
One day Grand Master Oogway had a vision that Tai Lung was going to escape and return to the valley of peace to get the Dragon Scroll. A day was selected for the Dragon Warrior to be picked on and all the other members of the Furious 5 showed their skills. When it came time for her turn. Oogway sense the Dragon Warrior.
When Oogway was choosing Po fell right in front of her. Tigress looked at Oogway asking in confusion whether he was pointing at her to which he says "him". Then everyone exclaims in shock when Oogway declares "The Universe has brought us the Dragon Warrior".
The entire rest of the day Shifu tried to intimidate him into leaving and at night Tigress said that if he respected Kung Fu "He would be gone by morning".
To her utter dismay he did not leave. Later that day when Po was receiving Acupuncture. Tigress went and told the story of Tai Lung and told of how she never received the love Shifu gave to him.
The next time you see her she is, with the others eating. They talk briefly about food. And then Shifu walks in revealing that Tai Lung has escaped and that Mast Oogway is dead and says that Po is the only hope to save the Valley despite Tigress saying they can.
Po then runs off to be stopped by Shifu. After Po decides to stay, Tigress disobeys Shifu's orders to go fight Tai Lung only to be followed by her fellow team mates in the coming battle.
While Po trains, Tigress and the 5 go to face Tai Lung and while they do work well together they ultimately fail and are defeated.
After Po finsihes training she was seen in defeated state. Also when Po finds out the thing is blank. Shifu sends the 5 and Po to evacuate the valley.
After Po defeats Tai Lung, Tigress shows respect bowing to him along with the rest of the 5.
Po then goes to see Shifu and that is how the 1st movie ends.