The abbey is the place where they all went to find refuge and protection. Its description says that it is "castlellated." This means that it was built up quite fancy and big enough to serve, as the word says, a castle. In those days, castles were the homes of the high end clergy and the aristocracy. In their castles, entire towns could be protected from attacks and other forms of danger. This abbey, however, was peculiar. There were turns and curves that made people look as if they disappeared. There is also a color represented in each room. It is said that these are colors of different stages of life, from birth (white) to death (black)
With this information, you can infer that the abbey in the story represents the false idea of omnipotence. The fake sense of protection, and isolation from danger that is caused by feelings of grandiosity and ignorance.
The abbey is a also a symbol of the inevitability of fate over life. The doors of the abbey would be shut to avoid entrance and exit. The place was supplied for, and prepared to hold balls and parties in the fantasy that they were safe there. In the end, the Red Death still penetrated the room with all its protection and pomp. Fate was to call the end of the story.