An octave is 8 days of commemoration of a major feast, including the day itself. The 8th day is called the octave day, and always falls on the same day of the week as the feast itself. So the octave day of Christmas is New Year's Day.
The first octave that was kept was the dedication of the Churches of Tyre and Jerusalem, under Constantine - these solemnities, in imitation of the dedication of the Jewish Temple, lasted eight days. In the fourth century, Easter and Pentecost were given octaves, and from this time onwards the celebration of octaves is becomes more frequent. In the sacramentaries of Gelasius and St. Gregory, on the octave day the office of the feast is repeated, but there was no provision for the intermediate days.
Amalarius tells us that it was customary in his time to celebrate the octaves of the feasts of SS. Peter and Paul and other saints. By the thirteenth century, perhaps under the influence of the Franciscans, octaves were extended to many other feasts.
In the Tridentine calendar of 1568, octaves were celebrated by the same office being repeated on each day of the octave. Some octaves overlapped, especially in the period after Christmas, so there were multiple commemorations of octaves.
In the calendar, as reformed by Pope St Pius X in 1911, there are:
1. Octaves of the first rank: Easter and Pentecost - no other feast may be celebrated during this time;
2. Octaves of the second rank: Epiphany and Corpus Christi - the octave days are greater doubles, and the days within the octave are semidoubles, being displaced only by doubles of the first class;
3. Octaves of the third rank: Christmas, Ascension, Sacred Heart - displaced by any feast day above rank of simple;
Immaculate Conception, Assumption, Nativity of Saint John the Baptist, Solemnity of St. Joseph, SS Peter and Paul, All Saints, and the principal patron saint of a church, cathedral, order, town, diocese, province, or nation - displaced by any feast day above rank of simple.
Simple octavesSaint Stephen, Saint John the Evangelist, Holy Innocents, Saint Lawrence, Nativity of Mary, and secondary patrons - kept as doubles of the second class, octave day was a simple, no days within the octave commemorated.
In later reforms, octaves have come under the knife, as have vigils, the alleged principle being for greater simplicity. In the reform of 1955, only the octaves of Christmas, Easter and Pentecost were kept, and during these octaves there are no commemorations.
In the Novus Ordo, the octave of Pentecost was suppressed. This appears to have been a mistake, and took Paul VI by surprise when, on Whit Monday, he appeared at his private chapel to celebrate mass, and was met with green vestments, rather than the accustomed red for the Holy Spirit. "But, it's Whit Monday." "You've abolished it, Your Holiness."
One octave which is not celebrated liturgically, but is often observed, is the "Octave of prayer for Christian unity," which runs from 18th January to 25th January - from St Peter's Chair at Rome to the Conversion of St Paul. This was started in 1908, approved by Pope St Pius X, and Pope Benedict XV encouraged its observance throughout the church.
The Catholic encyclopaedia on Octaves: http://www.newadvent.org/cathen/11204a.htm