The Dragon Warrior is a legendary prodigy described in the first film as someone who will know the secret behind the Dragon Scroll. By reading this scroll, the Dragon Warrior shall have "limitless power" in knowing the true secret to Kung Fu and also themselves. This "true secret" is later translated to mean that the prodigy will "know that the power is only within them, having to only believe in themselves". Grand Master Oogway understood this simple logic and thus, he created the Dragon Scroll, its reflective golden surface symbolizing this knowledge.
The identity of the Dragon Warrior was a mystery for a thousand years. Master Tigress, the leader of the Furious Five, shared a small piece of the legend behind the Dragon Warrior in the first Kung Fu Panda, saying that the warrior could "survive for months at a time on nothing but the dew of a single ginkgo leaf and the energy of the universe". It is unknown where this piece of the legend originated from, but Po heavily disagreed when he was told this, saying "[he]'ll need a lot more than dew and . . . 'universe juice'".
Master Shifu also shared a bit of legend behind the Dragon Scroll, saying that although no one truly knew what would happen when the Dragon Warrior read it, legend says "[they] will be able to hear a butterfly's wing beat . . . see light in the darkest cave, [and] . . . feel the universe in motion around [them]". This shares similarities with the legend behind the birth of Kung Fu, when Master Oogway could do such things after having achieved "excellence of self" ("Kung Fu"). This ties back into the "true secret" behind the scroll and the power of the Dragon Warrior. Even though it's possible, this may have been slightly misinterpreted, as many seemed to think it would hold a literal secret (in writing) to achieving limitless power.
In Kung Fu Panda, when Master Oogway received a vision of Tai Lung's return, he declared it was time to choose who will be the Dragon Warrior, the one who would stop Tai Lung and save the Valley of Peace. A tournament was held to determine which of the Furious Five would be given the title, and all of the Valley showed up to see who would be chosen, with Po declaring it as the "greatest day in Kung Fu history". But the choosing ceremony was interrupted when Po himself crash-landed in the arena in order to see the Dragon Warrior. Even though Master Shifu objected this was an accident, Oogway announced that the panda was the one, brought by the universe itself. Upon meeting Shifu, Po was told that he would never be the Dragon Warrior, until he uncovered the secret of the Dragon Scroll and mastered the highest level of Kung Fu. Throughout most of the movie, Shifu and the Five are far from convinced that Po was really the Dragon Warrior, but with his perseverance and eventual defeat of Tai Lung, they came to believe he was worthy and accepted him as a fellow Kung Fu master.
Years prior to the original movie, Shifu thought for a time that the Dragon Warrior would be Tai Lung, who went on a rampage when Oogway refused him the title (because he saw the true darkness in Tai Lung's heart, despite the love father and son shared). Tigress was also likely believed to become the Dragon Warrior, as Shifu protested to Oogway that Po fell in front of her before he was about to choose her.
In Kung Fu Panda Holiday, Shifu told Po that it was one of the most critical of the Dragon Warrior's duties, as the feast is very formal and the masters of all twenty-nine schools of Kung Fu were to be present. As the Dragon Warrior, Po was tasked with preparing for the event and seeing that every detail was just right, as well as learning its many customs.
In Kung Fu Panda 2, it is further assumed that the Dragon Warrior is considered as highly revered, as villager-turned-Kung Fu master Po seemed to have become something of a leader to the Furious Five, despite being less experienced. But when tasked with saving Kung Fu and China from Lord Shen's weapon, Po took on this challenge as his duty as the Dragon Warrior, despite his father's plea to not go on the dangerous journey. Throughout their adventure together, the Five assisted Po with his many blunders; however, they also followed his lead even in his seemingly less-than-brilliant plans. In the end, Po alone brought the defeat of Lord Shen and his weapon, ending with another victory for the Dragon Warrior and presumably silencing any remaining doubts of the Panda's worthiness of the title.