analytic geometry mostly studies geometric figures defined by linear and quadratic equations, in 2 or 3 dimensional affine space over the real numbers. algebraic geometry studies geometric loci defined by polynomials in any number of variables in affine or projective space of any dimension, over any field, as well as abstract versions of these loci defined analogously to manifolds by covering "charts", which themelves can be isomorphic to any affine locus. In particular "singular" points are welcomed, which are points where the locus is not like a manifold but can cross it self or have kinks and folds. In all these cases the functions acting on the loci are polynomials, or derived from them. In abstract algebraic geometry, an attempt is made to further include as rings of "functions" not just polynomials over a field, but any commutative ring with identity whatsoever. In this theory, one starts from such a ring A, and forms the set spec(A) consisting of all prime ideals of A. This is then given a topology in which the "closed" points are the maximal ideals, and prime ideals of coheight r are thought of as subloci of dimension r.
Over the complex number field, the study of geometric loci of dimension one in the "plane" i.e. C^2, or the projective plane and polynomials and rational functions defined on them, is essentially equivalent to the study of one dimensional complex manifolds and holomorphic and meromorphic fuunctions defined on them.
so yes, it starts out a little like analytic geometry, but then you raise the degree and the dimension, and you generalize to more abstract fields and even rings. and you tend not to entertain transcendental functions like e^x, or sin and cos. and although you can imitate differential calculus, it is harder to do integral calculus, although i suppose the complex analytic theory of residue, which you can imitate, gives you a hand in that direction.
as example, the ring R[X] where R = reals, gives a space spec(R[X]) consisting of all prime ideals of R[X], i.e. zero, and all ideals generated by irreducible linear or quadratic real poynomials. If C = complexes, then spec(C[X]) is zero and all ideals generated by linear polynomials X-z where z is a complex number. The ring inclusion R[X]-->C[X] induces by pullback a geometric map spec(C[X])-->spec(R[X]) that is generically 2 to 1, roughly with each pair of conjugate complex numbers mapping to the irreducible real quadratic with those roots, and branched over the "real line" consisting of the maximal ideals of R[X] with linear generators. So from this point of view, the space spec(R[X]) has more information than just the real solutions of real polynomials, it also incorporates Galois orbits of complex solutions. Thus the theory lends itself also to study of number theory.
There are some general analogies with linear algebra, but geared up. Just as one linear equation on k^n defines a linear subspace of codimension one, so (if we assume k algebraically closed) does one polynomial equation on k^n define an algebraic variety of codimension one. More generally, the codimension of the locus in k^n defined by r equations cannot be more than r, in the general case as well the linear case. A surjective linear map from k^m to k^n has all fibers as linear spaces of dimension m-n, while a surjective polynomial map k^m-->k^n has all fibers of dimension at least m-n, and the general one of exactly that dimension.
if you want to begin reading about algebraic geometry, and are really a beginner, a good book is Algebraic Curves, by Robert Walker, or maybe with a bit more algebraic background, Undergraduate algebraic geometry, by Miles Reid. A fantastic book is the huge, scholarly tome: Plane algebraic curves, by Brieskorn and Knorrer. Oh another excellent one is Riemann surfaces and algebraic curves, by Rick Miranda. Bill Fulton has made his lovely 1969 book on curves available for free:
http://www.math.lsa.umich.edu/~wfulton/CurveBook.pdf
Basic algebraic geometry, by Shafarevich, is an excellent introduction to higher dimensional algebraic geometry, i.e. not just curves. All these books are more introductory than Mumford's red book. Mumford's book is of course wonderful, but you will appreciate it more with some background from some of these other books, which have more examples and exercises, and are less abstract.
as a measure of the difference in analytic geometry and algebraic geometry, even in dimension one, note that every (projective) plane curve, over the complex numbers, is a compact surface. those studied in analytic geometry, namely circles, parabolas and hyperbolas, are all (over the complexes) just spheres, whereas those of higher degree are compact surfaces of arbitrary genus g ≥ 0. E.g. plane cubics have genus 1 and smooth plane quartics have genus 3. Indeed defining the genus was a primary contribution by Riemann to the study of plane curves.
The three main theorems about plane curves are the bezout theorem on the number of intersections of two plane curves, the resolution of singularities saying that every plane curve with singularities is the image by a degree one map of a curve having no singularities, and the riemann roch theorem which computes the number of rational functions on a given curve with a given set of poles. all three of these theorems are proved in Walker and Fulton.
generalizing these theorems to higher dimensions have been a primary focus of research for a 150 years or more. The general riemann roch theorem was proved by hirzebruch in the 1950's i think and generalized further by grothendieck in the 1960's. the bezout theorem has been beautifully generalized by fulton in his book Intersection theory, and the resolution of singularities was published by hironaka in 1964 in characteristic zero, and announced by him this year(!) in characteristic p.
http://www.math.harvard.edu/~hironaka/pRes.pdf
