I wold like to know what does it mean to say something is a set.
For me, any collection of elements is a set. Then this Russel paradox says collection of all sets is not a set.
All this came when looking at locally small categories. They are the categories where Hom class of any two objects is a set.
What is a set? What is a class? Do we loose anything if we assume all categories are locally small?
Set theory makes a distinction between sets and classes in order to avoid Russell's paradox and related inconsistencies. Briefly, given any property of sets, you can form the class of all sets with that property. You can not do the same for sets: there is only a set of sets with a given property if you can prove it from the axioms of set theory.
We can make an additional philosophical distinction between sets and classes: while both sets and classes can contain elements, only sets can be contained as elements of these. It is possibly enlightening to consider a third possibility: some frameworks for set theory allow one to discuss "ur elements" which can be contained by sets and classes, but cannot themselves contain elements. (I suppose there's also a fourth possibility of entities which can neither contain nor be contained, but then there's no way for them to interact with anything.)
If we're working in a set-theoretic framework, we need to make these distinctions if we want to be precise, because Russell's paradox is real. As a way of thinking, it is sometimes convenient to "ignore" the problem, at least temporarily, but only when one believes the details can be worked out later.
Finally, regarding your question about the harm in assuming all categories are locally small, my answer is an unsatisfying "it depends." The vast majority of categories you run across in practice seem to be locally small, for sure, so it's often a harmless assumption. In my opinion, however, it's probably best to keep the abstract idea of a "category" somewhat fluid, and use a suitable notion for whatever context you're working with. That notion may often be "locally small category," but I think it's a harm to make that a rigid definition to the point where you reject working with anything that doesn't fit it. This question on Math Overflow may interest you.
Mike Shulman's Set Theory for Category Theory goes into this in probably more detail than you want.
For ZFC, a class is the extent of a relation in the ambient first-order logic. In other words, there is a one-to-one correspondence between predicates $\varphi$ and classes. Every set gives rise to a class because we can always defined the predicate $\varphi(x)\equiv x \in S$ for a set $S$, but there are predicates, such as $\varphi(x)\equiv \top$, the class of all sets, for which there is no set, $X$, that can be defined in terms of the ZFC axioms such that $\forall x. x\in X \Leftrightarrow \varphi(x)$.
Classes are a metalogical concept in ZFC. NBG (von Neumann–Bernays–Gödel) set theory has an explicit sort of classes which allows classes to be directly manipulated.
The category of locally small categories, what we usually mean by $\mathbf{Cat}$ is not locally small. This is easy to show: consider the endofunctors of $\mathbf{Set}$, i.e. $\mathbf{Cat}(\mathbf{Set},\mathbf{Set})$. Just considering constant functors already produces a proper class as there's a constant functor for each set.
Sometimes as with the notion of a Grothendieck universe (and also in a slightly different way in algebraic set theory), we simply declare some sets to be the "small" sets. In particular, we declare a set (the universe) to be the set of all "small" sets. In this formulation, sets that aren't "small" play the role of proper classes. Since a Grothendieck universe is a model of ZF set theory, it can't exist within ZF set theory or else ZF set theory would be able to prove its own consistency. A more powerful notion of set theory is needed if you want to have a Grothendieck universe be a set. One option is Tarsk-Grothendieck set theory which states that each set is a "small" set for some Grothendieck universe. This is way more than is needed for most applications of Grothendieck universes but can be useful for higher category theory.
The universe idea can also be explored from a type theoretic context. This is a different notion of "universe", but it serves much the same purpose.
Well be prepared to have your mind blown. Not all "collections" can be called "sets" in a rigorous form of set theory, or else we run into paradoxes such as Russel's Paradox, where a set $A$ has itself as an element ($A \in A$), which unfortunately leads to so-called "naive set theory" (where every "collection" is a set) being inconsistent theory.
In set theory, this is typically resolved using the Zermelo-Frankel-Choice (ZFC) or Neumann-Berney-Godel (often called NBG) axioms. Basically, NBG is just a "conservative extension" of ZFC, and both treatments are equivalent. You can start reading about the ZFC axioms here and NBG axioms here (and the other answers provide even better references), but truly understanding such treatment involves having a pretty good handle on mathematical logic, which isn't necessarily required to understand small, large, and locally small categories.
Saunders Mac Lane (seen as generally the "standard" text of category theory) resolves the issue in a pretty informally simple way in Ch. I Sec. 6 (pg. 21-24) that is ultimately equivalent to ZFC, which I reference in my definition below. Essentially what is going on is we define the so-called "set of all sets" $U$ to be a collection with closure properties in a way where neither $U$ nor any other potentially problematic "collections" are included.
Definition. We define the collection $U$ as follows:
(i.) $x \in u \in U$ implies $x \in U$;
(ii.) $u, v \in U$ implies $\{u, v\}, \langle u, v \rangle, u \times v, u-v \in U$;
(iii.) $x \in U$ implies $\mathcal{P}(x), \bigcup x \in U$;
(iv.) $\mathbb{N} \in U$;
(v) if $f \colon a \to b$ is a surjective function with $a \in U$ and $b \subset U$, then $b \in U$;
A collection $a$ is a set if and only if $a \in U$ (note that these closure properties of $U$ are redundant, but sufficient as a criteria for determining which collections are sets).
A category $\mathcal{C}$ is small if $Obj(\mathcal{C}) \in U$, and large otherwise. Morever, $\mathcal{C}$ is locally small if for each $A, B \in Obj(\mathcal{C})$, we have $Hom(A, B) \in U$.
With the way this is set up, note that we can just talk about "small" and "large" categories in terms of naive set theory and get on with our lives. But I understand that this state of affairs leads to a bit of a predicament.
In particular, where you might be unsatisfied is the fact that I used "collections" and haven't exactly defined what those are, and that leads to answering second question, since a "class" is really what I've been calling a "collection" (and what category theory texts call "small sets") this whole time. I don't think you're ever going to be 100% happy (so much as 60% happy) with any definition you read of "classes" until you brush up on enough formal logic where you're comfortable looking at logic symbols. Because what separates "axiomatic set theory" from "naive set theory" is precisely the careful use of symbolic logic vs. treating any old collection with criteria $\{ x \colon \phi(x) \}$ as a set. There just isn't really a sufficient shortcut. But I'll do my best in terms of "informal logic" in the definition below.
Definition. A "class" is a logical statement $\Phi(x)$ with a variable $x$ (treated as any "set") in the "language of set theory". A given set $z$ is contained in a class if and only if the statement $\Phi(z)$ is true.
Such classes $\Phi$ define a set if there is some $z \in U$ such that $\forall x(x \in z \leftrightarrow \Phi(x)).$
Personally, I believe thinking of classes as simply logical statements that are separate and isolated from "sets" (as you'll find standard set theory texts often do more rigorously than I)--except when classes serve as legitimate "definitions" for particular sets--resolves a lot of annoying tension in your brain that distracts you from settling "more important" matters in mathematics.
The big thing to get is that it doesn't matter if the mathematical statements are regarded (in the "language" you're using) as sets or not. These "proper classes" can still be talked about as simply statements about sets. For example, the universal set $U$ can be denoted as the statement "$x=x$", which is always true for any set $z$. Another important example is the class $ON$ of ordinal numbers, where a set $x$ is an ordinal if and only if $\forall y (y \in x \rightarrow y \subset x) \land \exists z(z \in x \land \forall y(y \in x \rightarrow y \notin z))$. The class $Card$ of all cardinal numbers can be established similarly but defining the statement for it takes a quite a bit more work.
You can of course extend your "language of sets" to a "language of classes" (where the logical statements refer to classes instead of sets) and that way talking about classes is formalized "less awkwardly". This is the language that the NBG axioms operate on that conservatively extend ZFC in the language of sets. But you run into the same problems where you have statements about classes (such as again $x=x$ which is true for all classes) that can't be considered to define a "class", and we have the same issue all over again. That's why it's better to just appreciate the fact that everything you talk about in math can be written (in some weird way) in the language of sets and just move on with life.
And to answer your final (and arguably most important question in terms of category theory) we don't lose anything "important" to assume all categories are "locally small", since the goal of category theory is to generalize mappings. To do this, we need to be responsible in our distinction between "small" vs. "large" categories in category theory terminology (analogous to "sets" vs. proper "classes" in set theory terminology), so that we have $\mathbf{Set}$, $\mathbf{Grp}$, $\mathbf{Top}$, etc., with all the nice pretty functors between them, but without the paradox-related problems. However, you'll find not only $\mathbf{Set}$, $\mathbf{Grp}$, and $\mathbf{Top}$, but other large categories (such as $\mathbf{Rng}$, $\mathbf{K}$-$\mathbf{Vec}$, $\mathbf{Grph}$, $\mathbf{pTop}$, and even $\mathbf{Cat}$!) we care about are all "locally small". So we lose nothing in practical mathematics to conveniently assume that all categories are "locally small".
Sorry that it's a bit of a long post, and I hope this clears at least some of the confusion.
