The context, I am told, is that Hilbert wanted to illustrate that the uniqueness of prime factorization is not a completely obvious claim that needs no proof.
To this end, we introduce Hilbert numbers (which indeed have a rather silly definition out of context) and Hilbert primes (which are Hilbert numbers $n$ whose only Hilbert number divisors are $1$ and $n$ itself). Hilbert primes are not Hilbert numbers which are prime in the ordinary sense: for example, $9$ is a Hilbert prime because its only nontrivial factor, $3$, is not a Hilbert number.
Every Hilbert number has a "prime factorization" into Hilbert primes. But that factorization is not unique; for example, $441$ has two prime factorizations $$441 = 21 \cdot 21 = 9 \cdot 49.$$
(I'm not positive that Hilbert himself used this as an example; I can't find a source for this.)