What does it mean when I say that addition/multiplication for an equivalence relation is well defined?

Consider this equivalence relation on $\mathbb N$: $$a \sim b \quad\text{iff}\quad \lfloor a/10 \rfloor = \lfloor b/10 \rfloor $$ which says that two naturals are related if they differ only in their last digit.

This is a perfectly good equivalence relation, but we can't extend addition to equivalence classes by the same rule that works for modular arithmetic:

$$ [a]_\sim + [b]_\sim = [a+b]_\sim $$

The problem is that the sum of two equivalence classes now depend on which representatives we use to define their sum. For example, the rule seems to imply that $$ [11]_\sim + [32]_\sim = [43]_\sim \\ [17]_\sim + [35]_\sim = [52]_\sim $$ However $[11]_\sim$ is the same equivalence class as $[17]_\sim$, and $[32]_\sim = [35]_\sim$, but $[43]_\sim$ is not the same as $[52]_\sim$. So the definition doesn't really tell us which of the equivalence classes should be the result of $$ \{10,11,\ldots,19\} + \{30,31,\ldots,39\} $$

Checking that addition on the equivalence classes "is well-defined" means convincing oneself that this situation does not occur for the equivalence relation you're looking at.

Just because a relation is an equivalence, this doesn't mean it has to be "nice" with respect to any operation you'd like to put on its classes. To see this, look at a non-example of something being well-defined.

Let $V=\mathbb{R}^2$ be the plane with its usual vector space structure. Put a relation on $V$ by defining $u \sim v$ if $u = cv$ for some non-zero scalar $c$. This is an equivalence relation on $V$. The zero vector is in a class by itself, and the other classes are the vectors that form parallel lines through the origin (with the zero vector removed).

Since classes are naturally represented by vectors, you could attempt to define a natural addition on $V/\sim$ by $$ [u] + [v] = [u+v]. $$ But this fails to be well-defined, even though the relation in play is an equivalence.

The problem is that two different people could add the SAME TWO classes but get different answers. That's no good. For example, Alice wants to add the class corresponding to the $x$-axis (with zero removed) to itself. This class is $[(1,0)]$, for example. So, Alice does $$[(1,0)] + [(1,0)] = [(2,0)] = [(1,0)].$$

Bob will do the exact same class addition, but recognize that $[(1,0)]$ is also equal to $[(-1,0)]$. Then Bob gets $$[(1,0)] + [(-1,0)] = [(0,0)] \neq [(1,0)].$$

Alice and Bob have added the exact same class to itself, but got conflicting answers.

Suppose we have partitioned the set of all integers into $\mathbb Z_{10}$, the set of all equivalence classes modulo $10$. Let $[n] = \{x \in \mathbb Z : x \equiv n \pmod{10} \}$ be the equivalence classes.

Now let's define $[a] \vee [b] = [\operatorname{lcm}(a, b)]$ for all $[a], [b] \in \mathbb Z_{10}$

Then \begin{array}{c} [2] &\vee &[6] &= &[6] \\ [12] &\vee &[16] &= &[48] \\ \end{array}

But $[2]=[12], \ [6]=[16]$ and $[6] \ne [48]$. It follows that $\vee$ is not well defined on $\mathbb Z_{10}$.