To add to what Charles wrote, another reference is Mac Lane and Moerdijk's Sheaves in Geometry and Logic. They prove something a bit more general, involving Lawvere-Tierney topologies on a topos. For the purposes of understanding what I'm about to write, it's not necessary to know what a Lawvere-Tierney topology is.

Mac Lane and Moerdijk's book contains the following two results:

1. Let $\mathcal{E}$ be a topos. Then the subtoposes of $\mathcal{E}$ (i.e. the reflective full subcategories with left exact reflectors) correspond canonically to the Lawvere-Tierney topologies on $\mathcal{E}$.

2. Let $\mathbf{C}$ be a small category. Then the Lawvere-Tierney topologies on $\mathbf{Set}^{\mathbf{C}^{\mathrm{op}}}$ correspond canonically to the Grothendieck topologies on $\mathbf{C}$.

Result 1 is almost part of Corollary VII.4.7. The "almost" is because they don't go the whole way in proving the one-to-one correspondence, but I guess it's not too hard to finish it off. (Edit: it also appears as Theorem A.4.4.8 of Johnstone's Sketches of an Elephant, where Lawvere-Tierney topologies are called local operators.) Result 2 is Theorem V.4.1.

I agree with the point of view that Charles advocates. When I started learning topos theory I got bogged down in detailed stuff about Grothendieck topologies, and it all seemed pretty technical and unappealing. It wasn't until years later that I learned the wonderful fact that Charles mentions: an elementary topos is Grothendieck iff it's a subtopos of some presheaf topos. I wish someone had told me that in the first place!

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To add to what Charles wrote, another reference is Mac Lane and Moerdijk's Sheaves in Geometry and Logic. They prove something a bit more general, involving Lawvere-Tierney topologies on a topos. For the purposes of understanding what I'm about to write, it's not necessary to know what a Lawvere-Tierney topology is.

Mac Lane and Moerdijk's book contains the following two results:

1. Let $\mathcal{E}$ be a topos. Then the subtoposes of $\mathcal{E}$ (i.e. the reflective full subcategories with left exact reflectors) correspond canonically to the Lawvere-Tierney topologies on $\mathcal{E}$.

2. Let $\mathbf{C}$ be a small category. Then the Lawvere-Tierney topologies on $\mathbf{Set}^{\mathbf{C}^{\mathrm{op}}}$ correspond canonically to the Grothendieck topologies on $\mathbf{C}$.

Result 1 is almost part of Corollary VII.4.7. The "almost" is because they don't go the whole way in proving the one-to-one correspondence, but I guess it's not too hard to finish it off. Result 2 is Theorem V.4.1.

I agree with the point of view that Charles advocates. When I started learning topos theory I got bogged down in detailed stuff about Grothendieck topologies, and it all seemed pretty technical and unappealing. It wasn't until years later that I learned the wonderful fact that Charles mentions: an elementary topos is Grothendieck iff it's a subtopos of some presheaf topos. I wish someone had told me that in the first place!