Categories for the Working Mathematicians says
In addition to the metacategory of all sets ~ which is not a set ~ we want an actual category $Set$, the category of all small sets. We shall assume that there is a big enough set $U$, the "universe", then describe a set $x$ as "small" if it is a member of the universe, and take $Set$ to be the category whose set $U$ of objects is the set of all small sets, with arrows all functions from one small set to another.
and
we may consider the metacategory of all categories: its objects are all categories, its arrows are all functors with the composition above. Similarly, we may form the category $Cat$ of all small categories - but not the category of all categories.
and
7. Large Categories
In many relevant examples, a category consists of all (small) Mathematical objects with a given structure, with arrows all the functions which preserve that structure. We list useful such examples with their monics.
What do "large" and "small" sets mean and how do they differ from each other?
What do the metacategory of all sets and $Set$ mean and how do they differ from each other? What does it mean by the metacategory of all sets is not a set, and is $Set` a set?
What do "large" and "small" categories mean and how do they differ from each other?
What do the metacategory of all categories and the category $Cat$ of all small categories mean and how do they differ from each other?
Thanks.
1) You gave the definition of that. The set of all sets is not a set (Russell's paradox), but we want the collection of objects of a category to be a set. Thus we cannot define the category of sets with all sets as objects. What we do here instead is restrict the sets that we consider (actually you will not really feel any restriction; it is just a formality). There is the notion of a Grothendieck universe which is basically just a big set of sets that is closed under pretty much everything you want to do with sets. If we fix such a universe $\mathcal{U}$, we can consider all sets that are contained in that universe (these sets are called small or $\mathcal{U}$-small then) and the collection of these will be a set again. Thus we can define the category of small sets (with respect to a Grothendieck universe). If a set is not small, we call it large and that is the difference. It will not be considered in that category, but one can actually enlargen the given universe to be able to still talk about all sets of interest. This has to do with strongly inaccessible cardinals.
2) Since Maclane wants his collections of objects to be sets (one could also take classes) he needs a name for the situation when these form a class (a proper class). That is what he calls metacategory. The difference was already explained in 1).
3) If one defines categories with classes (for objects and morphisms), then one differs between large, locally small and small categories. A category is called small if the collections of objects and of all the morphisms between objects are sets. If only the latter is true, a category is called locally small. If we deal with proper classes for both the objects and the morphisms, we call the category large.
4) Same as for sets. See the answers above.
To summarize: It is a formality and you can basically still think of these categories of containing everything that you want.