 For an abelian variety "A", the Albanese variety is "A" itself, so the dual should be "Pic"0("A"), the connected component of what in contemporary terminology is the Picard scheme.For the case of the Jacobian variety "J" of a compact Riemann surface "C", the choice of a principal polarization of "J" gives rise to an identification of "J" with its own Picard variety.This in a sense is just a consequence of Abel's theorem.