If I have a charged body near a conducting cylinder, am I supposed to used Poisson's or Laplace's equation? I know that I am supposed to use Laplace's equation only when the total charge density is zero, but I was thinking if I have the description of the electric field, I could do it just like the case of a sphere in a constant electric field where we can use Laplace's equation. I don't understand when to use Poisson's equation rather than Laplace's.
Always use Poisson's equation. That is the general formula that will hold in E&M (in the classical Maxwell formalism). However, it will simplify to Laplace's equation if you are trying to solve the Poisson equation in a region of space where there is no net charge density at any point.