This is a very broad question, we have a huge numbers of such characterization.
But part C of "Sketches of an elephant" contains most of those I know. Basically all the notion introduced have such a "site characterization", i.e. a properties of a topos or a morphism is characterized by the existence of a Site description having some properties.
And I would like to add that Part C is in my opinion the better written and easiest to read part of the elephant.
From memory, you can find their at the very least conditions for:
- Atomic toposes, as sheaves for a topology where all non-empty sieve are covering (called an atomic topology).
- Proper geometric morphism from a 'finite subcovering property'
- Coherent toposes by combining the fact that every covering is finitely generated with existence of finite limits in the site.
and a lots of other examples (open geometric morphisms, locally connected morphisms and so one...) but I do not remember all of them.