Is there a way to characterize (e.g. by their local components) the cuspidal automorphic representations that arise from elliptic curves? There is a procedure to go from elliptic curves to cuspidal automorphic representations of $GL_2$ but can the image be characterized?
I'll assume that you're talking about elliptic curves over $\mathbf{Q}$  much of what I'm going to say should generalize to totally real fields, replacing modular forms by Hilbert modular forms. I think the question is much less wellunderstood outside of that setting.
So your question is a little bit backwards: there's a procedure to attach an elliptic curve to certain automorphic representations of $\mathrm{GL}_2/\mathbf{Q}$, and the modularity theorem says that this is surjective. The automorphic representations in question are exactly those spanned by a Hecke eigenform of weight 2 with Hecke eigenvalues contained in $\mathbf{Q}$.
Let $f = \sum_n a_n q^n$, $a_n \in \mathbf{Q}$ be such an eigenform, normalized with $a_1 = 1$. We can think of $f$ as a function on $\mathrm{GL}_2(\mathbf{Q}) \backslash \mathrm{GL}_2(\mathbf{A}_\mathbf{Q})$, and it spans an irreducible cuspidal automorphic representation $\pi_f$ of $\mathrm{GL}_2$. (This transformation is described in many places, such as here or here.)
We can decompose $\pi_f = \bigotimes'_p \pi_p$. The fact that $f$ is a modular form of weight $2$ corresponds to $\pi_\infty$ being the discrete series representation of $\mathrm{GL}_2(\mathbf{R})$ of weight $2$.
For $p$ not dividing the level of $f$, we have that $\pi_p$ is the unramified principal series representation of $\mathrm{GL}_2(\mathbf{Q}_p)$ with Satake parameters equal to the two roots of the polynomial $x^2  a_p x + p$.
It's a bit harder to characterize $\pi_p$ for $p$ dividing the level  these will be ramified representations of $\mathrm{GL}_2(\mathbf{Q}_p)$. However, the strong multiplicity one theorem says that the automorphic representation is determined by all but finitely many of its components anyway.
Now, given such a modular form $f$, we can construct the corresponding elliptic curve over $\mathbf{Q}$ as follows. We regard $f$ as a oneform on the modular curve $X(N)$. The Hecke operators away from $N$ act by correspondences on $X(N)$, so they determine a legitimate commuting action on its Jacobian $J(N)$. The Jacobian $J(N)$ breaks up into a product of "Hecke eigenspaces", and the elliptic curve $E_f$ is the quotient of $J(N)$ on which the Hecke operators act by the $a_p$'s.
What happens when we relax some of the conditions we put on $f$? If we do not require the Hecke eigenvalues $a_p$ to be in $\mathbf{Q}$, the same construction produces an abelian variety of dimension greater than $1$. The corresponding Galois representation will be a $2$dimensional representation over the field $K$ generated by the $a_p$'s, and the abelian variety will have "real multiplication" over $K$.
If we replace $f$ by a Hecke eigenform of higher (even) weight, the corresponding motive looks like a symmetric product of elliptic curves. On the level of Galois representations, a weight $2k$ Hecke eigenform with rational eigenvalues corresponds to a $2$dimensional representation over $\mathbf{Q}$ with "HodgeTate weights" $(k, 0)$. Roughly speaking, this means it looks like $\mathrm{H}^k(X, \underline{\mathbf{Q}_\ell})$ for a smooth proper variety $X$.

1$\begingroup$ Over other number fields, there will be a similar conjectural description of which ones come from elliptic curves, but it won’t be probable. $\endgroup$ – Will Sawin Jul 21 at 9:26