It is said to have appeared first in the late work of Riemann, and was extensively studied by Wirtinger in 1895, including degenerate cases.
Given a non-constant morphism of algebraic curves, write Ji for the Jacobian variety of Ci.
To qualify that somewhat, to get an abelian variety, the connected component of the identity of the reduced scheme underlying the kernel may be intended.
The theory of Prym varieties was dormant for a long time, until revived by David Mumford around 1970.
One advantage of the method is that it allows one to apply the theory of curves to the study of a wider class of abelian varieties than Jacobians.