In Ullman's Introduction to Automata, Languages and Computation (1979)
Let $M = (Q, \Sigma, \delta, q_0, F)$ be a DFA. Relation $R_M$ is defined as: for any $x$ and $y$ in $\Sigma^*$, let $xR_My$ if and only if $\delta(q_0, x) = \delta(q_0, y)$.
An equivalence relation $R$ such that $xRy$ implies $xzRyz$ is said to be right invariant (with respect to concatenation). We see that every finite automaton induces a right invariant equivalence relation.
Is a right invariant equivalent relation necessarily $R_M$ for some finite automaton $M$?
Theorem 3.9 (The Myhill-Nerode theorem). The following three statements are equivalent:
The set $L \subseteq \Sigma^*$ is accepted by some finite automaton.
$L$ is the union of some of the equivalence classes of a right invariant equivalence relation over $\Sigma^*$ of finite index.
Let equivalence relation $R_L$ over $\Sigma^*$ be defined by: $x R_L y$ if and only if for all $z \in \Sigma^*$, $xz$ is in $L$ exactly when $yz$ is in $L$. Then $R_L$ is of finite index.
Suppose $L$ is a regular language. If $L$ is the union of some of the equivalence classes of a right invariant equivalence relation $R$ over $\Sigma^*$ of finite index, is there some finite machine $M$ s.t. $L$ is the language which $M$ accepts and $R=R_M$?
My questions above come from having difficulty understanding what the Myhill-Nerode theorem means.