A complex torus $X$ of complex dimension $1$ is just a quotient of $\mathbb{C}$ by a lattice $\Lambda=\mathbb{Z}\oplus \tau\mathbb{Z}$ where $\tau=a+ib$ is a complex number with $b>0$. The complex structure is determined by this $\tau$. I want to undestand this complex structure as an endomorphism $J:TX \to TX$ such that $J^2=-1$ (an almost complex structure). I feel like, regarding X as a Lie group, it is a invariant complex structure. If so, $J$ is determined by a $2 \times 2$ matrix.
Now, for a $2\times 2$ matrix $J$ satisfy $J^2=-1$, it must be something like $$J=\left(\begin{array}{cc} \frac{\alpha}{\beta} & \frac{1}{\beta}\\ -\frac{\alpha^2}{\beta}-\beta & -\frac{\alpha}{\beta} \end{array}\right). $$
What are the relations between $\tau=a+ib$ and the coefficients $\alpha, \beta$?
The number $\tau$ is a global constant associated to your torus $X$, and is (essentially) uniquely determined by the "complex structure" of $X$ if you insist on $|\tau|\geq1$, $0\leq{\rm Re}(\tau)\leq{1\over2}$. This number $\tau$, e.g., governs the length of closed geodesics on $X$.
Contrasting this the matrix $J$ is a purely local object that somehow encodes the CR equations of the admissible conformal parameters $z$.