The feast celebrates the primacy of Peter and his successors' the Pope. The great sculpture (see left) of the chair of St. Peter houses the wood fragments of a chair believed to be St. Peter's. The chair seems to float in midair supported by four great doctors of the Church, Sts Augustine and Ambrose of the West and Sts. Athanasius and John Chrysostom of the East. On this feast day the chair is covered in hundreds of candles. If you click on the photo you might be able to see them.
The other tradition for today's feast is the statue of St. Peter within the basilica is dressed in regal vestments. The interesting feature of this particular statue is that his foot is almost worn away by the number of pilgrims who rub it each day.