39 pages, LaTeX2e, no figures; new proofs added in Version nr. 2, some arguments rewritten and typos corrected. Final version to appear in Journal of Geometry and Physics.We define a Fourier-Mukai transform for a triple consisting of two holomorphic vector bundles over an elliptic curve and a homomorphism between them. We prove that in some cases the transform preserves the natural stability condition for a triple. We also define a Nahm transform for solutions to natural gauge-theoretic equations on a triple -- vortices -- and explore some of its basic properties. Our approach combines direct methods with dimensional reduction techniques, relating triples over a curve with vector bundles over the product of the curve with the complex projective line.The authors are members of VBAC (Vector bundles on algebraic curves), which is partially supported by EAGER (EC FP5 Contract no. HPRN-CT-2000-00099) and by EDGE (EC FP5 Contract no. HPRN-CT-2000-00101). This research has been partially supported by the Italian/Spain bilateral programme Azione Integrata, IT203 ”Sheaves on Calabi-Yau manifolds and applications to integrable systems and string theory” and by the research projects BFM2003-00097 of the Spanish DGI and SA118/03 of the “Junta de Castilla y León”.Peer reviewed