To translate his result for people not familiar, unique factorization means that a number uniquely decomposes into primes. You almost certainly learned this happened in school for integers (Z), but it does not apply to all cases.
Quadratic Numbers in Algebraic Number Theory terms are Q[sqrt(-d)], that is, a+b\sqrt(-d) where a and b are rational numbers. d=5 is the first number we can pick where unique factorization does not hold.
In fact, the Stark-Heegner theorem tells us something even more powerful: if d is squarefree, the only imaginary quadratic fields containing unique factorization are when d=1, 2, 3, 7, 11, 19, 43, 67, and 163. Any other choice (or any choice containing a square of any prime, e.g. 4=2^2), and unique factorization will fail.
I've left out what a prime, or indeed an irreducible, mean in this case, but what's astounding at least to me is that there are only 9 such numbers where it works, and this is provable. Heegner did that aged 50+.