The twelve solemnities the cathedral lists are Epiphany (January 6), Easter Sunday, the Ascension, the Apparition of the Apostle at Clavijo (May 23), Pentecost, the Martyrdom of St. James (July 25), the Assumption (August 15), All Saints (November 1), Christ the King, the Immaculate Conception (December 8), Christmas (December 25), and the Translation of the Apostle's remains (December 30). On those days it is swung during the entrance procession or at the end of Mass.
On any other day, seeing it depends on whether a pilgrimage has asked for it. Groups write to botafumeiro@catedraldesantiago.es. The cathedral's pages state no fee for the request and publish no list of the days on which it has been requested, so a pilgrim arriving on an ordinary day cannot know in advance. While the restoration notice stands, do not plan a journey around it; look at the cathedral's liturgy page shortly before you travel.
The Botafumeiro is the cathedral's great censer. It weighs 53 kg (117 lb), stands 1.5 m (about 5 ft) tall, hangs from some 20 m (65 ft) up, and is swung by eight men called tiraboleiros. The one in ordinary use is brass, made by José Losada in 1851; a silver replica was given in 1971. When it does swing, the cathedral fills early, and backpacks are not allowed inside, so leave yours behind and arrive well before Mass.