Also known as Dies Sanctissimi Corporis et Sanguinis Domini Iesu Christi (Day of the Most Holy Body and Blood of Jesus Christ the Lord) or the Solemnity of the Most Holy Body and Blood of Christ, the Solemnity celebrates the Real Presence of the Body and Blood, Soul and Divinity of Jesus Christ in the Eucharist.
Saint Juliana of Liege worked tirelessly for nearly forty years of work for the Institution of the Feast (see here for more on both Saint Juliana and her work on the Feast). Pope Urban IV instituted the feast for the entire Church as a Solemnity in 1264 by the papal bull Transiturus de hoc mundo. Though Holy Thursday commemorates the Institution of the Eucharist, the liturgy also commemorates Christ's washing of the disciples feet, the institution of the priesthood and the agony in the Garden. Pope Urban, agreeing with Saint Juliana, decided that with all of the other functions taking place on Holy Thursday the principal event (the Eucharist) was often overlooked or lost sight of.
When Pope Pius revised the Calendar, the Feast of Corpus Christi was one of only two "feasts of Devotion" that he kept (the other was Trinity Sunday). He also set the date for the Thursday following Trinity Sunday. The day is one of five total occasions in which a diocesan bishop is not to be away from his diocese unless for a grave or urgent reason. In many countries the Feast of Corpus Christi is considered an holy day of obligation - in countries where it is not the date is generally moved to the following Sunday. Traditionally a procession is held with prayers and singing to honor the Blessed Sacrament as it is held aloft.
In many countries there are specific folk festivals or celebrations held, unique to that country's culture. In England, the feast day was a time for mystery plays, especially during the medieval ages until the feast was suppressed during the protestant reformation. Corpus Christi is celebrated in Catalonia with a tradition of a dancing egg. This tradition dates from at least the early 16th century.