Buddha Purnima, also widely known as the Buddha Jayanti celebrates the birth of Lord Buddha across the globe. Traditionally, it's a holiday in Mahayana Buddhism commemorating the birth of the prince Siddhartha Gautama, later the Gautama Buddha—founder of Buddhism.
According to many beliefs, Lord Buddha was born in Lumbini, which is now Nepal, around 563 BCE. The legend has it that when Prince Siddhartha Gautam was born in a royal lineage, an astrologer predicted that either he will set to become a great king else he would denounce everything and leave the material things.
Naturally, prince's father king Suddhona wanted his son to reign as a powerful and greatest of kings, therefore he did not let him go out of the palace walls. However, as destiny would have it—prince at the young age of 29, went out of the palatial comforts and saw how people suffer, and how difficult it is for an ordinary man to survive.
According to Buddhist literature, it is believed that what prince witnessed during this time is known as the four sights. He got to understand the suffering of a common man, saw an old man, a sick person, a dead body and finally an ascetic holy man, who was content in life.
It is believed that these experiences were vital enough and compelled the prince to leave his comfortable royal life behind and move ahead to embrace divinity by walking on the spiritual path, leading to enlightenment.
You can chant Lord Buddha's mantra "Namu Myōhō Renge Kyō" which is chanted within all forms of Nichiren Buddhism as well as Tendai Buddhism. It is the Lotus Sutra which helps in attaining mental peace and people believe it cures them of all problems.