I want to have an overview of all the meaningful equivalence relations defined on $X^{\ast}$, in particular when the languages in question are regular. They typically arise in connection with the minimization of automata. Further I want to know all the relations among them. Below I write what I know, I encourage everyone to add useful facts to this list, give other equivalence relations and how they relate to the given ones!?Common equivalence relations on $X^{\ast}$ are\begin{align*} u \equiv_L v & :\Leftrightarrow  \forall w \in X^{\ast} : uw \in L \leftrightarrow vw \in L \\ u \equiv_R v & :\Leftrightarrow  \forall w \in X^{\ast} : wu \in L \leftrightarrow wv \in L \\ u \equiv_S v & :\Leftrightarrow   \forall w_1, w_2 \in X^{\ast} : w_1uw_2 \in L \leftrightarrow w_1vw_2 \in L\end{align*}If we have some deterministic and complete automaton $\mathcal A = (X,Q,\delta,q_0,F)$ with $L = L(\mathcal A)$, then every $u \in X^{\ast}$corresponds to a function $\varphi_u : Q \to Q$. Then we can define$$ u \equiv_A v :\Leftrightarrow \delta(q_0, u) = \delta(q_0,v) \Leftrightarrow \varphi_u(q_0) = \varphi_v(q_0).$$Also for states $q, q' \in Q$ we could define$$ q \equiv q' :\Leftrightarrow \forall u \in X^{\ast} : \delta(q, u) = \delta(q',u) \Leftrightarrow \varphi_u(q) = \varphi_v(q')$$and by incorporating the final states (and thereby fitting it more to the conrete language $L$) we could define$$ q \equiv^L q' :\Leftrightarrow \forall u \in X^{\ast} : \delta(q, u) \in F \leftrightarrow \delta(q', u) \in F.$$And if we have some recognizing monoid $M$ with homomorphism $\varphi : X^{\ast} \to M$ we could define$$ u \equiv_h v :\Leftrightarrow h(u) = h(v).$$We have (the automaton is arbitrary)\begin{align*}  u \equiv_L v & \Leftrightarrow \delta(q_0, u) \equiv^L \delta(q_0,v) \\ u \equiv_R v & \Leftrightarrow \forall q \in Q : \delta(q, u) \in F \leftrightarrow \delta(q, v) \in F \\ u \equiv_S v & \Leftrightarrow \forall q \in Q : \delta(q, u) \equiv^L \delta(q,v) \\ u \equiv_h v & \Rightarrow u \equiv_S v \\ u \equiv_S v & \Rightarrow u \equiv_L v \land u \equiv_R v \\ u \equiv_A v & \Rightarrow u \equiv_L v \\ \varphi_u = \varphi_v & \Rightarrow u \equiv_S v \\ u \equiv_S v & \Leftrightarrow \forall q \in Q : \varphi_u(q) \in F \leftrightarrow \varphi_v(q) \in F\end{align*}And we have that the syntactic monoid is precisely the transformation monoid of the minimal automaton (which could be constructed with $\equiv_L$). Also the above notions could be described in terms of language quotients $u^{-1}L := \{ w : uw \in L \}, Lu^{-1} := \{ w : wu \in L \}$ (and the minimal automaton could also be constructed with the quotients $u^{-1}L$) or the sets $L_q(\mathcal A) := (X,Q,\delta,q,F)$ (i.e. the language accepted if $\mathcal A$ is started at $q$).
Useful equivalence relations on $X^{\ast}$ (like the Nerode and syntactic equivalence relations)
fl.formal languages;automata theory;regular language
The article by Thomas Colcombet on Green's Relations and Their Use in Automata Theory might be interesting for you. These relations are very common in the algebra of semigroups. See also Section 2.8 in Chapter 10 of the Handbook of Formal Languages.