One-dimensional complex tori are just elliptic curves and are all algebraic, but Riemann discovered that most complex tori of dimension 2 are not algebraic via the Riemann bilinear relations.
Essentially, these are conditions on the parameter space of period matrices for complex tori which define an algebraic subvariety.
Finding criteria for a complex torus of dimension 2 to be a product of two elliptic curves (up to isogeny) was a popular subject of study in the nineteenth century.
The surface is diffeomorphic to S1×S1×S1×S1 so the fundamental group is Z4.
Hodge diamond: Examples: A product of two elliptic curves.