The category of schemes has a large (and to me, slightly bewildering) number of what seem like different Grothendieck (pre)topologies. Zariski, ok, I get. Etale, that's alright, I think. Nisnevich? pff, not a chance. There are various ideas about stacks I would like to test out, but the sites I am most familiar with have few application-rich topologies. (Smooth, finite-dimensional manifolds are particularly boring in this respect, and topological spaces are not much better)
What I'm after is a table listing the well-known/common topologies on $Sch$ and their relative 'fineness'. Or, if you like, containment. We of course have the canonical topology – is there a characterisation of that in terms of schematic properties, as opposed to the obvious categorical definition?
And furthermore, one expects that for nice schemes, various topologies will coalesce, say one sort of covers becoming cofinal in another, when restricted to a subcategory of $Sch$. Say those schemes which are Noetherian, smooth or even just varieties.
Then there are things like when categories of sheaves, or 2-categories of stacks, are equivalent. But maybe this is asking too much.
Maybe I'm after something like 'Counterexamples in Grothendieck topologies'. Does such a thing exist, all in one place? I'm sure it is all there in SGA, or the stacks project, or in Vakil's Foundations of Algebraic Geometry, but I'm after the distilled essence.
PS I am interested in things which are (pre)topologies even if they are not usually used as such for the purposes of sheaves.
EDIT: I'm not merely after examples of Grothendieck topologies on $Sch$, even though that is handy. I want a reference, if there is one, or just a straight-out answer, that compares the various topologies on $Sch$, and under which circumstances (restricting $Sch$ to a subcategory) they coincide.
For example, does an fppf cover of a variety have local sections over an etale cover? Do the fppf and fpqc topologies give rise to the same sheaves over a nicely behaved scheme? Is the etale topology strictly 'weaker' than some other topology no matter what schemes one looks at? Does one get the same Deligne-Mumford stacks for topology A and topology B?
(Grumble over)
Figure 1 on page 7 of these notes gives a few more of the less common (pre)topologies: cdp, fps$\ell'$ etc.
Best Answer
The basic answer is essentially as Emerton described in the comment. The most commonly used topologies on schemes are Zariski, Nisnevich, étale, smooth, syntomic, fppf, and fpqc, and this list is totally ordered by increasing fineness. The canonical topology is finer than the fpqc topology, but I have never seen it explicitly used. You can see a discussion of these topologies (other than Nisnevich) in the Stacks project chapter on Topologies on Schemes.
You ask about restricting to subcategories of schemes to get equivalent topologies, but I think you would have to take unusually small subcategories. For example, the étale and Nisnevich topologies coincide on the spectra of fields only when the fields are separably closed. I think if the Nisnevich covers of a scheme are Zariski covers, then the scheme is zero dimensional. Smooth and étale covers coincide if you restrict to say, varieties of a single fixed dimension. I think the same is true for syntomic versus smooth and fppf versus symtomic (but I am far from sure). If you restrict your schemes to be locally finitely presented over a fixed base, then fppf and fpqc coincide. Even though the étale and smooth topologies are usually not equivalent, they give rise to equivalent categories of sheaves, because every smooth cover has an étale refinement.
The Stacks project has a list of properties that different topologies satisfy, in the Descent chapter. Bjorn Poonen also has a table of permanence properties in Appendix C of his notes on Rational points on varieties.
If you're really hoping for a more interesting looking partially ordered set of topologies, you may consider more exotic examples like the cdh topology (finer than Nisnevich, incomparable with étale), and the naïve fpqc topology, whose covers are faithfully flat quasi-compact maps (incomparable with most of the list). The latter is typically only used by people when they are making mistakes.