On that same day, the mother would be ritually purified (thus the "purification").
Saint Mary and Saint Joseph kept this law, even though, since Saint Mary remained a virgin after the birth of Christ, she would not have had to go through ritual purification. In his gospel, Luke recounts the story (Luke 2:22-39).
When Christ was presented in the temple, "there was a man in Jerusalem named Simeon, and this man was just and devout, waiting for the consolation of Israel" (Luke 2:25) When Saint Mary and Saint Joseph brought Christ to the temple, Simeon embraced the Child and prayed the Canticle of Simeon:
Now thou dost dismiss thy servant, O Lord, according to thy word in peace; because my eyes have seen thy salvation, which thou hast prepared before the face of all peoples: a light to the revelation of the Gentiles, and the glory of thy people Israel (Luke 2:29-32).
Originally, the feast was celebrated on February 14, the 40th day after Epiphany (January 6), because Christmas wasn't yet celebrated as its own feast, and so the Nativity, Epiphany, the Baptism of the Lord (Theophany), and the feast celebrating Christ's first miracle at the wedding in Cana were all celebrated on the same day. By the last quarter of the fourth century, however, the Church at Rome had begun to celebrate the Nativity on December 25, so the Feast of the Presentation was moved to February 2, 40 days later.
Inspired by the words of the Canticle of Simeon ("a light to the revelation of the Gentiles"), by the 11th century, the custom had developed in the West of blessing candles on the Feast of the Presentation. The candles were then lit, and a procession took place through the darkened church while the Canticle of Simeon was sung. Because of this, the feast also became known as Candlemas. While the procession and blessing of the candles is not often performed in the United States today, Candlemas is still an important feast in many European countries.
This emphasis on light, as well as the timing of the feast, falling as it does in the last weeks of winter, led to another, secular holiday celebrated in the United States on the same date: Groundhog Day. You can learn more about the connection between the religious holiday and the secular one in Why Did the Groundhog See His Shadow?