Abelian varieties are higher dimensional analogues of elliptic curves (which are included) – they are varieties equipped with a structure of an abelian group, hence abeliangroup schemes, whose multiplication and inverse are regular maps.
Goro Shimura, Abelian varieties with complex multiplication and modular functions, Princeton Univ. Press 1997# David Mumford, Abelian varieties, Oxford Univ. Press 1970
Alexander Polishchuk, Abelian varieties, theta functions and the Fourier transform, Cambridge Univ. Press 2003
M. Demazure?, P. Gabriel, Groupes algebriques, tome 1 (later volumes never appeared), Mason and Cie, Paris 1970 – has functor of points point of view; for review see Bull. London Math. Soc. (1980) 12 (6): 476-478, doi
J. S. Milne, Abelian varieties, course notes, pdf
Daniel Huybrechts, Fourier-Mukai transforms in algebraic geometry, Oxford Mathematical Monographs. 2006. 307 pages.
André Weil, Courbes algébriques et variétés abéliennes, Paris: Hermann 1971
C. Bartocci, Ugo Bruzzo, D. Hernandez Ruiperez, Fourier-Mukai and Nahm transforms in geometry and mathematical physics, Progress in Mathematics 276, Birkhauser 2009.