the difference between a round table and a normal table is that a round table has no ends. therefore, you can arbitrarily select which seat is to be called seat #1.
two ways to approach this problem:

(1)
you can fix one of the people in place (or, equivalently, rotate the table like a carousel so that that person always winds up in the same place). this will avoid the production of multiple equivalent scenarios in which the people are seated in the same order, but have just shuffled a seat or two over (these don't count as different arrangements).
then you have free rein to arrange the other four people, so that's 4! = 24 arrangements.

(2)
alternatively, you can figure out that there are 5! = 120 arrangements overall.
however, each UNIQUE arrangement is actually repeated five times: because there are five seats at the table, there are 5 different versions of every possible seating arrangement. (for instance, ABCDE, BCDEA, CDEAB, DEABC, EABCD are all equivalent.)
so this means you must divide by 5: 120 / 5 = 24