Pidyon haben is a ceremony that recognizes the first born male child (to be specific, the first born male child that was born naturally). The ceremony arose due to the special status of the firstborn in biblical society. Traditionally, the firstborn of Israel were supposed to be dedicated to God (Exodus 13:1-2), and expected to perform religious services for the priests (Kohanim). According to the biblical command, they could be redeemed with five shekels. This redemption is the purpose of pidyon haben.
A pidyon haben is required on the 30th day after the birth of a first born male child. Today, the child is typically released from his obligations by the payment of five dollars in coins.
If the father is a Levite or a Kohen, (making the child a Levi or a Kohen) the pidyon haben does not apply (logically, since the father could wind up paying himself). If the mother is the daughter of a Levite or a Kohen the child is exempt.
Pidyon haben is observed in traditional communities, and in the conservative community. It tends not to be observed by Reform Jews.
Sources: Shamash and Joseph Telushkin. Jewish Literacy. NY: William Morrow and Co., 1991.