Grothendieck topology for a non-small category To define a Grothendieck topology of a category, we usually require that the category is small. 
Question 1: Why do we need to require the category to be small?
I thought that the problem was that we wanted a sieve to be a set. (In the category of manifolds, the maximal sieve of an arbitrary manifold $X$ is not a set.) But my thought must be wrong because of the following remark.
When we want to deal with the category of manifolds (schemes, or topological spaces), Metzler says that we can avoid this problem by choosing a subcategory which is small. For example, the category $\mathbf{M}$ of smooth manifolds which are second countable and Hausdorff.
Question 2: Why does this trick work?
Metzler explains the reason in Remark 13 in the linked paper, but I can not understand it (because I do not know what is the problem.)
Question 3: If we define a Grothendieck topology as an equivalence class of basis of Grothendieck topology, does this definition work?
In the definition of a basis of Grothendieck topology (See Def 5 in the paper by Metzler for the definition), we seem to deal with only sets. So I think that the definition is well-defined for any category.
 A: Re 2:
In my opinion, none of the answers just yet have hit the nail on the head about "why this trick works". The real reason is Urs' comment about dense subsites. Whether one takes manifolds to mean 2nd countable + Hausdorff, or whether one removes these conditions and considers all topological spaces with a smooth atlas, the topos of sheaves over their corresponding sites are equivalent. This is why the trick works; it's not simply that you don't care about pathological manifolds (it's actually very convenient to have them around when you're talking about differentiable stacks), but that you don't need to include them in your site of definition, because they are faithfully represented by the sheaves they induce over any full subcategory of manifolds, containing at least one manifold of each dimension. E.g., one can consider the full subcategory of manifolds spanned by only those of the form $\mathbb{R}^n,$ and takes sheaves on this site, and this is equivalent to taking sheaves on the site of all manifolds. The latter site is NOT essentially small, but it doesn't matter. In full generality, you need a site with a "small set of topological generators": see SGA4 (in particular expose ii, theorem 3.4) and the link Urs posted http://ncatlab.org/nlab/show/dense+sub-site
A: The other answers are all good, but I thought I would also point out that one doesn't have to require that sites be small, or have small dense sub-sites, or satisfy WISC.  I think one does generally want to assume that each covering sieve contains one that is generated by a small family, and also an additional "solution-set condition" that is vacuous for small sites, which I studied in this paper.
In particular, I showed there that there is a reasonable notion of "the category of (small) sheaves" on such a large site, and in most cases it does admit a "sheafification" functor from the category of (small) presheaves.  What's different is that sheafification doesn't necessarily have a right adjoint, so that "sheaves" can't necessarily be identified with particular presheaves.  The "category of small sheaves" is generally not a Grothendieck topos, nor an elementary one, but it satisfies all of Giraud's axioms except for the existence of a small generating set, and it has "the same" universal property (in a certain sense) as the topos of sheaves on a small site.
A: Ad 2: The chapter on set theory in the Stacks Project explains in detail how to construct small categories of schemes, closed under various operations, and then the chapter on topologies constructs the usual topologies (Zariski, fpqc, fppf, étale, etc.) on these categories - without running into the set-theoretic difficulties which all(?) the other sources ignore. The rough idea is to use the reflection principle from mathematical logic in order to get a large enough ordinal number $\alpha$ which bounds the schemes under consideration. In practice, I think that you can ignore $\alpha$ because you can always choose it large enough, at least as for the proofs of specific theorems for example about cohomology groups etc.
As David has explained, we don't have to do all this in the case of manifolds, which probably was the case the question was aiming at. Therefore this answer here is just additional info for the interested reader...
A: *

*We need the category $C$ to be small so as to define the functor category $Cat(C^{op},Set)$, i.e. the category of presheaves, so that it is locally small.

*The trick works with the category of manifolds because the category of smooth Hausdorff second countable manifolds is equivalent to a small category (use the embedding theorem, so that every such manifold is embedded in a high-dimensional $\mathbb{R}^n$ - there are only a set of such submanifolds if we take them to be literal subsets of Euclidean spaces). People generally don't want to deal with non-Hausdorff manifolds (some do) and almost never non-second countable ones (only show up as pathological counterexamples, for example the long ray). So one can repeat this trick for any large category which is equivalent to a small category.

*There might be a proper class of bases for any given topology on a large site.
It is entirely equivalent to work with a pretopology/basis rather than a topology, as one doesn't need the full topology to define sheaves (or possibly stacks etc), which is the whole point. Once you have the topos of sheaves, the choice of basis is immaterial, though sometimes convenient for calculations.
The real thing you want is that the axiom WISC is satisfied, especially when the definition of a pretopology (=basis) doesn't demand that each object only has a set's worth of covering families. Consider for example the category of groups with the pretopology consisting of single epimorphisms as covering families. This is a large category such that each object has a proper class of covering families. But one can find a set of epimorphisms such that any epimorphism is refined by one in that set, and this is the statement of WISC. For a small category, WISC is automatically satisfied, and also when the definition of basis is such that only a set of covering families is associated to any object.
A non-example is the category of schemes with the fpqc topology, and so there we cannot sheafify an arbitrary presheaf. Another non-example can be the category of sets if one assumes the negation of the axiom of choice (independent work of van den Berg, Karagila and myself).
A: 
Over a large site there is in general no sheafification functor
  https://ncatlab.org/nlab/show/sheafification. For how to get around
  this, the keyword is dense subsite
  https://ncatlab.org/nlab/show/dense+sub-site.

The above content is a comment by Urs to the answer of David Roberts. I thought it needs to be in the form of an answer. So, made it a CW post.  
