In this article we study a Rankin-Selberg convolution of two complex variables attached to Siegel modular forms of degree 2. We establish its basic analytic properties, find its singular curves and compute some of its residues. In particular, we show that two known Dirichlet series of Rankin-Selberg type, one introduced by Maass and another by Kohnen and Skoruppa, are obtained as residues from this series of two variables. Furthermore, we define and study a collection of Rankin-Selberg convolutions for Jacobi forms of degree 1.