We should limit to books which we can really recommend, either by their special content, approach or pedagogical value. Historically fine but outdated books are in a separate historical section below. The outstanding surveys may be added to the lists if they are not too specialized to minor directions: the subfields may be covered in separate lists. See also MathOverflow discussions best-algebraic-geometry-text-book-other-than-hartshorne, life-after-hartshorne-the-book
A. 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 (listed also under scheme theory); for review see Bull. London Math. Soc. (1980) 12 (6): 476-478, doi
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.
P. Griffiths, J. Harris, Principles of algebraic geometry
Phillip A. Griffiths, Introduction to algebraic curves
Daniel Huybrechts, Complex geometry - an introduction, Springer (2004). Universitext. 309 pages.
Mori program and birational geometry
János Kollár, Shigefumi Mori, Birational geometry of algebraic varieties, With the collaboration of C. H. Clemens and A. Corti. Translated from the 1998 Japanese original. Cambridge Tracts in Math. 134 (1998), viii+254 pp.
Kenji Matsuki, Introduction to the Mori program, Universitext. Springer 2002. xxiv+478 pp. MR2002m:14011
Herbert Clemens, János Kollár, Shigefumi Mori, Higher-dimensional complex geometry, Astérisque 166 (1988), 144 pp. (1989).
Goro Shimura, Abelian varieties with complex multiplication and modular functions, Princeton Univ. Press 1997
Dale Husemöller, Elliptic curves, Graduate Texts in Mathematics. 111 (2nd ed.). Springer 2004, ISBN 0-387-95490-2.
Anthony Knapp, Elliptic curves, Math Notes. 40. Princeton University Press 1992
Neal Koblitz, Introduction to elliptic curves and modular forms, Graduate Texts in Mathematics. 97 (2nd ed.). Springer-Verlag 1993
Joseph H. Silverman, The Arithmetic of Elliptic Curves, Graduate Texts in Math. 106, Springer 1986; Advanced Topics in the Arithmetic of Elliptic Curves Graduate Texts in Mathl. 151, Springer 1994.
J. H. Silverman, John Tate, Rational Points on Elliptic Curves, Springer 1992.
Gerd Faltings, Lectures on arithmetic Riemann-Roch theorem, Annals of Math. Studies 127, Princeton Univ. Press 1992
S. Bosch, W. Lütkebohmert, M. Raynaud, Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer 1990. x+325 pp. MR91i:14034
Cohomology theories for schemes
J. S. Milne, Etale cohomology, Princeton Univ. Press 1980, gbooks
P. Berthelot, A. Ogus, Notes on crystalline cohomology, Princeton Univ. Press 1978. vi+243, ISBN0-691-08218-9
something basic on log schemes, e.g. from Kato/Ogus/Olsson
Algorithmic and computational methods
Things like Groebner bases, combinatorical methods with toric varieties etc.
David A. Cox, John B. Little, Don O’Shea, Ideals, varieties, and algorithms
Historically important but now outdated
While many of these fine books are still pleasure for some readers, they do not capture the modern viewpoint and have maybe too old notation to be regularly used. But one should be aware of them, and of sometimes unique material exposed there.
W. V. D. Hodge, Daniel Pedoe, Methods of algebraic geometry, 3 vols. (see review by Coxeter in Bull. Amer. Math. Soc. 55, 3, part 1 (1949), 315-316, euclid)
F. Severi, Vol. I (1942): Serie, sistemi d’equivalenza e correspondenze algebriche sulle varieta algebriche. Vol. I I (1958) and I I I (1959): Geometria dei sistemi algebrici sopra una superficie e sopra una varieta algebrica.