“Sets,” “classes,” and “collections”: what’s the difference? After reading about Russel’s Paradox and the Burali-Forti paradox, I’ve been doing some research into how classes work.
On Wikipedia, a class is defined loosely as

a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share.

What exactly does “collection” mean here? I was under the impression that the purpose of formally defining sets was to formalize the intuitive notion of a “collection” - but the existence of classes that aren’t sets implies that sets don’t actually formalize our notion of a “collection” adequately.
If $\Omega$ (say) is the class of all sets, and classes are forbidden from containing themselves, we might define $\Omega’$ as the “collection” of all classes, such that $\Omega’$ is a type of collection that is not a class (call it a meta-class). Similarly, we can define the meta-meta-class $\Omega’’$ as the class of all meta-classes, and so on... eventually we can define $\Omega^n$ for any finite number $n$, and $\Omega^\alpha$ for any ordinal $\alpha$, and even define $\Omega^\Omega$ as the “collection” of all objects of the form $\Omega^\alpha$, where $\alpha$ is an ordinal. And so on.
Basically, we have to keep defining new “higher” types of containers to contain previous types of containers, in order to avoid paradox. But all of these things are apparently still “collections” (intuitively). Does it make sense to talk about a “collection of all collections,” or does this result in a paradox?
So my question is: what exactly does “collection” mean? Why is it so hard to formalize our intuitive notion of a “collection”, and why have sets and classes failed? What properties do “collections” have that sets and classes (and meta-classes, and meta-meta-classes, etc) don’t have?
 A: At some point in any system of mathematical formalism, you must have a definition that is intuitive rather than formal, just like you can't expect to have an English dictionary that defines every word without circularity. A "collection" is exactly what we expect it to be: it's what you think when you think the word "collection". "Sets" and "(proper) classes" are particular kinds of collection, distinguished by the formalism we brought in to try to make "collection" precise.
As for why it's so hard: Well, because our intuitive notion of "collection" is self-contradictory, by Russell's proof. The issue is that we want to say that a "collection" is not just a collection, but also a thing that could be placed in a collection. That seems intuitively reasonable, but it turns out it's not. So it's not that "set" and "class" fail to formalize a sensible idea; it's that "set" and "class" are the best formalizations we've found for an idea that's essentially nonsense.
Defining $\Omega'$ as the "collection of all classes" is deeply suspect. The issue is that a proper class is not really a thing on its own. We can talk about proper classes as if they were things, but only because it's a sort of "shorthand" for talking instead about the formulas that define them; if we allow a proper class to be considered a "thing" that might (for example) be represented as a variable, that exposes us to Russell's paradox again.
As for your last question: the property that "collections" have that formalized types of collections don't is that they're intuitive. That means they have all the freedom of the English language -- in particular, membership in a collection can be ambiguous. I can, for example, describe the collection "all of the collections which do not contain themselves", leaving this collection's membership in itself ambiguous. If you were to develop set theory in an environment where membership is allowed to be ambiguous (which has been done!) then you might find the notion of a "set" closer to your intuitive notion of a "collection".
