One has the notion of a compact Riemann surface with "many automorphisms", a concept for which there exist many equivalent definitions. You have correctly identified the "genus one Riemann sufaces with many automorphisms" -- a situation which is slightly complicated by the fact that, strictly speaking, the automorphism group of any complex elliptic curve as a Riemann surface is infinite -- so let's concentrate on the case of Riemann surfaces $X$ of genus $g \geq 2$, in which case the automorphism group is always finite.

The following three conditions on $X$ are equivalent:

(1) On the moduli space $\mathcal{M}_g$ of genus $g$ complex algebraic algebraic curves, the function $C \mapsto \# \operatorname{Aut}(C)$ has a strict local maximum at $X$ -- explicitly, there exists a neighborhood of $U$ of $X$ such that every curve in $U \setminus \{X\}$ has smaller automorphism group than $X$.

(2) The natural map $X \rightarrow X/\operatorname{Aut}(X)$ is a Belyi map -- i.e., the quotient has genus zero and the map is ramified only over three points. By the "easy" direction of Belyi's theorem, it follows that $X$ can be defined over some number field, and it is interesting to study the field of moduli of such curves $X$ (which in this case is known to be the minimal field of definition).

(3) $X$ is uniformized by a finite index torsion free subgroup $\Gamma$ of a hyperbolic triangle group $\Delta(a,b,c)$. This makes clear that these curves are a significant generalization of the Hurwitz curves, which correspond to the "first" hyperbolic triangle group $\Delta(2,3,7)$. At times I have expressed the opinion that it is strange that there has been so much research done on the $\Delta(2,3,7)$ case -- i.e., when the automorphism group is numerically as large as it could possibly be -- rather than on the more general case of $\Delta(a,b,c)$, i.e., the case where the automorphism group is sufficiently large to have the above very interesting analytic / topological / arithmetic implications.

For more information on these Riemann surfaces, see the bibliography of

http://math.uga.edu/~pete/triangle-091309.pdf

Especially highly recommended is [50], a paper by J. Wolfart. In general Wolfart is one of the pioneers of this subject and has written many interesting papers, mostly from the perspective of complex function theory.