What are some examples of classes that are not sets? After reading about Russell's paradox, I see that the set of all sets does not exist, so instead it is called a class.

What other commonly known classes exist that are not sets? I know the class of all singleton sets is not a set, because you can unite that class to get the class of all sets. This seems like the class of all sets of a fixed finite size is not a set either, correct? 

What are some examples of other 'things' that can't be gathered up and put into a set?
First, any description which doesn't uniquely specify the elements of a set tends to give a proper class.

For example, a group is an set together with operations satisfying a certain collection of axioms.  Since these axioms don't pin down the elements of the group, one might expect the collection of all groups to not be a set.

This holds in a similar fasion for topological spaces, rings, vector spaces, fields, manifolds, metric spaces, and many of the other commonly studied objects.

This also applies to your "collection of all singletons" example, and similarly to any sort of "collection of all sets of a fixed size".

But just because a description DOES pin down the elements doesn't mean you have a set.  For example, the collection of all ordinal numbers is not a set, despite the fact that if there is an order preserving bijection between two ordinals $\alpha$ and $\beta$, then one must have $\alpha = \beta$ as sets.

Similarly, the collection of all cardinals doesn't form a set.
In two-sorted first order set theory with proper classes (such as NBG) it is an axiom:
$$
\forall x\, \exists X\, \forall z\, [ z \in x \Leftrightarrow z \in X ]
$$
where lower-case letters denote type-0 objects (sets) and upper-case letters denote type-1 objects (classes).

This would mean that there are coclasses for every set:
$$
\forall x\, \exists \bar X \, \forall z\, [ z \in x \Leftrightarrow z \notin \bar X\, ]
$$
Thus, the elements NOT in a given set, forms a proper class.