Random function "returns a uniformly distributed int". Does this mean the probability of every number is the same?
Uniform distribution is when all values have the same probability. A random uniformly distributed integer in the range $0,\ldots,9$ will attain each of the values with probability $1/10$. In practice, you function is not truly random but only pseudorandom, so the probabilities won't be exactly $1/10$ but only very close to $1/10$.

Normal distribution is a probability distribution on real numbers. If we "bin" them then we get a normal distribution on the integers. If we "cap" it (there are several ways of doing this) then we get a probability distribution on a finite set of integers.