What does the symbol "$\mapsto$" mean in the logical expression "$J \vDash_{E[x \mapsto a]} (\alpha \implies \beta)$"? 
Consider an interpretation $j$ and an element $a$ $\in$ $\text{dom} (J)$. We also have that $$J \vDash_{E[x \mapsto a]} (\alpha \implies \beta)$$ where $\alpha$ and $\beta$ are formulas.

But I am not sure what this $\mapsto$ mean in this context?
 A: "$x\mapsto a$" means "the object $x$ is sent to the object $a$." This is often used to clarify the definition of a function, e.g. $$f:\mathbb{N}\rightarrow\mathbb{N}: n\mapsto 2^n+n.$$
In this case, though, I believe it refers to a variable assignment, saying that $x$ will get interpreted as $a$. E.g. under the valuation $x\mapsto 2$, the formula $x>1+1+1$ is false in $\mathbb{N}$. Remember that in order to evaluate the truth of a formula in a structure, we need to interpret each of its free variables; this variable interpretation is the subscript of the "$\models$" relation.
A: $x \mapsto y$ is the notation typically used to indicate that an element $x$ (from some domain of a function) is mapped to an element $y$ (frome some codomain of a function) by that function.
E.g., the successor function would be notated by
$$f: \mathbb{N} \to \mathbb{N} : x \mapsto x + 1 $$
meaning that the object $x$ – an element of $\mathbb{N}$ (the domain of $f$) – is being mapped to the object denoted by $x + 1$ – an element of $\mathbb{N}$ (the codomain of $f$).  
In your case, the function that is considered – $E$ – is an assignment function
$$E: VAR \to D$$
where $VAR$ is the set of variables and $D$ is the domain of the model, and $E_{[x \mapsto a]}$ is a variant assignment or $x$-alternative of $E$.
 A variant assignment of $E$ is an assignment that is just like $E$ except that it maps the variable $x$ to the object $a$ (instead of to whatever $E$ mapped it to).
Formally: For $x \in VAR$ and $a \in D$, the function $E_{[x \mapsto a]}$ is that assignment function $F$ such that
$$F(y) = \begin{cases}E(y) & y \not \equiv x\\a & y \equiv x\end{cases}$$
i.e., the variant assignment $F = E_{[x \mapsto a]}$ maps the same objects as $E$ does to all variables other than $x$, and possibly differs from $E$ in that it maps $x$ to the object $a$.
Note that $F$ can also be identical to $E$ in case that $E$ istself already mapped $x$ to $a$. 
Applied to the whole expression you gave, the notation means that the formula $(\alpha \to \beta)$ is valid under the interpretation $J$ and the variable assignment that is just like $E$ except that the variable $x$ is assigned the object $a$.
