Jacobian varieties are the most important class of abelian varieties.
To every nonsingular algebraic curve (over the complex numbers) of genus one associates the Jacobian variety or simply Jacobian either via differential 1-forms or equivalently via line bundles: Jacobian is the the moduli space of degree line bundles over , i.e. the connected component of the identity of the Picard group of .
The Abel-Jacobi map is defined with help of periods.
(one should also cover Abel-Jacobi theorem)