Definition of smallest equivalence relation
I’m going to guess that the actual context was more like the smallest equivalence relation on $A$ satisfying such-and-so conditions. If that’s the case, what is intended is the intersection of all equivalence relations on $A$ satisfying the given conditions: the intersection of equivalence relations on $A$ is an equivalence relation on $A$, and this intersection is the smallest subset of $A\times A$ (smallest in the sense of $\subseteq$) that is an equivalence relation on $A$ and satisfies the given conditions.
An equivalence relation is a set of ordered pairs, and one set can be a subset of another.
For any set $S$ the smallest equivalence relation is the one that contains all the pairs $(s,s)$ for $s \in S$. It has to have those to be reflexive, and any other equivalence relation must have those. The largest equivalence relation is the set of all pairs $(s,t)$.
For some in between examples, consider the set of integers. The equivalence relation "has the same parity as" is in between the smallest and the largest relations.
Think about how the relations "is congruent to mod $n$" are related by inclusion.
As @JiK comments, the equivalence relations get their "less than" order from the natural way that sets have such an order. That order is "partial" since there are pairs of equivalence relations such that neither is a subset of the other.
If you know the theorem that says that equivalence relations naturally correspond to partitions, you can translate the order structure. The partition $P$ is finer than the partition $Q$ if every block of $P$ is completely contained in some block of $Q$. Finer partitions correspond to smaller equivalence relations. In the finest partition every element of $S$ is in a block by itself - the smallest equivalence relation. In the coarsest partition all of $S$ is one block - the largest equivalence relation.
Equivalence relations are (partially) ordered by implication; $\Theta \leq \Phi$ if and only if $$ x \Theta y \implies x \Phi y $$ is an identity.
In fact, this partial ordering is a complete lattice; the meet operation (a.k.a. the greatest lower bound) is given by logical and. That is, the equivalence relation $\Theta = \bigwedge_{i \in I} \Theta_i$ is the one defined by
$$x \Theta y \Longleftrightarrow \bigwedge_{i \in I} x \Theta_i y$$
or in terms of quantifiers rather than infinitary operations, $$x \Theta y \Longleftrightarrow \forall i \in I: x \Theta_i y $$
If you look at the graph of the relations, this can all be rephrased in terms of subsets and intersections.
A simple source of examples is to use the first isomorphism theorem — there is a bijective correspondence between congruences and quotients.
e.g. in terms of sets, quotients can be viewed as partitions of the set, and the correspondence is that the equivalence classes of an equivalence relation are the parts of the partition.
The smallest equivalence relation on a nonempty set $X$ corresponds to the partition whose classes are all singletons; it is equality. The largest equivalence relation is the one with just a single partition; the corresponding equivalence relation is the one where everything is related. The ordering on equivalence relations corresponds to whether one partition refines another.