The clock in Poe's story is a symbol of death, reminding the party goers that their time is short.
The people in the castle are there trying to escape the "red death," a disease which sounds ominously like the Black Death or bubonic plague that wiped out a good portion of Europe's population in the 1300s and beyond.
The guests are partying, trying to enjoy life and forget the troubles surrounding them, when the clock chimes the hour with "brazen lungs." The sound is so loud and peculiar that it stops the orchestra from playing. People stop dancing and grow nervous and pale at this reminder of the passage of time and their own mortality. People pass their hands over their brows—everyone is briefly disconcerted until the chiming ends. Then the party resumes.
Poe devotes a whole paragraph to the chiming of the clock to demonstrate that even in this supposed sanctuary, the privileged few cannot escape the specter of death.