It is possible for discrete data to be modeled by a normal distribution.
If we toss a few coins and count the number of heads we get a binomial distribution which depends on the number of coins tossed. If we toss a thousand coins and count heads, the distribution we get will still be a binomial distribution. But we can model it as a normal distribution, with 500 heads as the mean and the usual bell shape around it.
It's not really a normal distribution. If we zoomed in we'd still see it's not possible to get 500.5 heads as it would be in a true normal distribution. But the normal distribution is a good approximation to the binomial distribution for big N.