Griffiths looks at algebraic varieties, always over the complex numbers, through the lens of complex manifold theory. His main tools are drawn largely from complex analysis of several variables. In that book he and Harris first develop foundational results for complex manifolds such as the Kodaira vanishing theorem, and the Hodge decomposition, and then apply them to the study of complex projective varieties. They also employ topological tools like Poincare duality, and later introduce and apply spectral sequences. The discussion includes some theorems generally included within the realm of differential gometry, such as a generalized version of the Gauss Bonnet theorem, apparently in the version due to Chern, a famous complex differential geometer. There is also some discussion of the Hirzebruch Riemann Roch theorem. Griffiths and his school are primary contributors to the field of Hodge theory, the study of cohomology of manifolds, especially algebraic ones, via the decomposition of their cohomology by harmonic differential forms. He has several seminal papers on periods of integrals, generalizing theories of Riemann and Abel and Torelli. Here is a link to his ICM talk from 1970 describing some of these ideas and their origins in the analytic theory of curves.
http://www.mathunion.org/ICM/ICM1970.1/Main/icm1970.1.0113.0120.ocr.pdf
I myself would say his work is within algebraic geometry because it studies algebraic geometric objects, but would consider the methods as differential analytic. I.e. I think of complex algebraic geometers as tending to study complex algebraic varieties with any methods available, algebraic, analytic, or topological. This was apparently the path pioneered by Abel and Riemann. By contrast, for a treatment of algebraic curves using only algebraic methods, such as integral ring extensions, see the book by William Fulton linked above (post 47).
I realize now that I have been somewhat cavalier about what context I am working in from time to time. Here is one of my papers in which it is stated that the field can be any algebraically closed one of characteristic ≠ 2, hence all methods must be algebraic.
http://alpha.math.uga.edu/%7Eroy/sv5rst2.pdf
and here is one where the field is restricted to the complex numbers:
http://alpha.math.uga.edu/%7Eroy/sv2rst.pdf
Here is another where the field must be the complex numbers, but that is not even stated.
http://alpha.math.uga.edu/%7Eroy/sv1nr.ps
Note also that Griffiths, in the 1970 talk linked above, speaks only of algebraic geometry, no mention of complex algebraic geometry, yet he immediately begins to write down complex path integrals.
I just noticed I myself wrote a brief essay "introducing" algebraic geometry to a class of graduate students taking the course, in case someone may get something from it:
http://alpha.math.uga.edu/%7Eroy/introAG.pdf
By way of disclosure, Griffiths is my mathematical "grandfather", in the sense that he advised my thesis adviser, C.H. Clemens.
