What precisely is lost when considering proper classes rather than sets? Motivated by concerns over the foundational issues vis-a-vis category theory. What is the essential useful characteristic of sets that is lost when instead considering proper classes? 
Referring to Hungerford's Algebra, it would seem that the notions of functions and surjection/injection carry over more or less identically. So what precisely is lost when we consider, for example, categories that are not locally small?
I understand totally the necessity of defining proper classes, that is to defined sets that would otherwise be self-referential in terms of membership.
 A: Some things that are lost:


*

*Arbitrary unions: The union of a class of sets may not be a set.

*Arbitrary unions/intersections/products: Much care is required to reframe things in some cases, because you can't, say, take $\bigcup_{i\in\Bbb N}\{C_i\}$ where the $C_i$ are proper classes.

*Power sets: The class of all subsets of a proper class is not a complete lattice.

*Pairing: If $A$ and $B$ are proper classes, there's no such thing as $\{A,B\}$

*Induction: You can't use well-founded induction to prove things about all classes as you can to prove things about all sets.

*Choice: The usual axiom of choice applies only to sets, not classes; adopting the axiom of global choice instead produces a strictly stronger theory.


There may be others.
A: Classes are essentially shorthands for formulas. You can work with a class, with some care, if you can write it as $\{x:\phi(x)\}$. So clases have to be rather precisely specified.
Now, there a lot of things you can do with sets you cannot do with classes:
Suppose you have two classes $\{x:\phi(x)\}$ and $\{y:\psi(y)\}$. You can also have a formula $F(x,y)$ such that for each $x$ such that $\phi(x)$ there is a unique $y$ such that $\psi(y)$ and $F(x,y)$. Essentially, $F$ is a class function between the given two classes. You can also express that $F$ is surjective. But this does of course not imply that there is a formula $G$ such that for all $y$ with $\psi(y)$ there is a unique $x$ with $\phi(x)$ and such that $F(x,y)$. So with classes, you can essentially have a surjection without a right inverse.
Suppose you have again two classes $\{x:\phi(x)\}$ and $\{x:\psi(x)\}$. Then you can form the intersection $\{x:(\phi\wedge\psi)(x)\}$. But if you have a sequence of classes parametrized by $n$ of the form $\{x:\phi_n(x)\}$, you cannot form their intersection, $\{x:(\phi_1\wedge\phi_2\wedge\ldots)(x)\}$ since the usual logic does not allow for infinite conjunctions $\phi_1\wedge\phi_2\wedge\ldots$
So there are a lot of things one can do with classes and one can make reasonable statements about categories that are not locally small. But it requires a deeper understanding of logic and set theory and there are important limitations. A very terse but good introduction to handling classes is given in the new edition of Kunen's Set theory.
A: Assuming that the framework is $\sf ZFC$, sets are actual objects which exist. Classes, namely proper classes, do not exist. Those are definable collections which we can talk about in the meta theory. In other words, we can talk about basic finitary operations about classes, but we can't really do much more.
What examples are there?


*

*Classes do not have power sets. We can't talk about the collection of all subclasses of a proper class.

*Classes are not objects, so they are not elements of other classes. For example, $\{V\}$ is not even a definable object, where $V$ is the class of all sets.

*Infinite collections of classes are highly limited. Classes are formulas with variables. This means that in order to talk about an infinite collection of classes, you need to have some formula which defines all the classes using different parameters. But if your classes are not uniformly definable, you can't talk about such collection.


There are other examples, some of which we can circumvent using clever tricks, others we can't. But the important point is that in the context of $\sf ZFC$ sets exists and proper classes don't. If we want our objects to actually exist in the universe then we need something which assures that they are sets, e.g. universes.
A: If you know a little set theory, there is a certain mental image that you can use to understand the difference between sets and classes.  Suppose that $\kappa$ is an inaccessible cardinal. Then we can make a model of set theory, called $V_k$, which consists of all sets whose rank is strictly less than $\kappa$. The subsets of this model are precisely the sets whose rank is strictly less than $\kappa+1$. The subsets of rank exactly $\kappa$ are proper classes from the point of view of $V_\kappa$. One can show that $V_k$ is a model of ZFC and $V_{\kappa + 1}$ is a model of Morse-Kelley set theory.
Now, if we have a set $a$ in this model (so $a \in V_\kappa$), we can do a lot of things to $a$ while staying in $V_\kappa$. We can take the powerset of $a$, and the powerset of that, and so on. We can even iterate the powerset $\alpha$ times for any ordinal $\alpha \in \kappa$, and we will stay in $V_\kappa$. 
This is not the case if we take a "proper class" $b$ of rank $\kappa$ in our model. We cannot take even one powerset, or do any other operation that increases the rank. So for example $\{V_{\kappa}\}$ itself, which has rank $\kappa + 1$. The answer by dfeuer has more examples. 
Now, the mental picture to have is that the entire universe of set theory is exactly like this, but viewed "from the inside" instead of being viewed from the outside as above. The class of ordinals is analogous to a large inaccessible cardinal $\Omega$. The objects of rank less than $\Omega$ are sets. The objects of rank exactly $\Omega$ (that is, the objects whose elements are all of rank $<\Omega$) are the proper classes. With sets, we can do many things and still have sets; with proper classes, we cannot do anything that increases the rank, because from our "inside" perspective the ordinals end at $\Omega$ (although we can easily imagine the ordinal $\Omega + 1$ and many more "small" extensions of $\Omega$). 
So the arguments that will go through for classes are the ones that do not rely on objects of larger rank. On the other hand, there are results for sets that necessarily use objects of much larger rank; the most famous of these if probably the Borel determinacy theorem. 
