>>10905191Classical algebraic geometry is (almost) geometry from high school. It's better to work over C than R, since C is algebraically closed. The algebraic geometry of Spec C[x1,...,xn] is more or less the same thing as describing shapes ('varieties') in C^n which arise as the simultaneous solution set to some finite family of complex polynomials in n variables. Built into this are the rational functions defined on certain subsets, and functions between varieties which can be described locally using rational functions.
Of course, we also care about real algebraic varieties (since antiquity), and number theorists care about solutions to (families of multivariate) polynomials over Z or Q or some other ring of integers or number field. Or if you care about infinitesimals, you might be interested in power series rings k[[x]] over a ring k. Schemes abstract this further so that the space only has to be locally describable in terms of zeroes of equations over rings (polynomials or not). This includes all of the above cases as well as projective versions of them.
Many common geometric constructions over complex or real varieties can be abstracted to any scheme, such as (a) localizations ("rational functions") defined on certain subsets, (b) tangent spaces, (c) cohomology (having to do with global geometric structure), (d) multiplicity, (e) critical points, (f) line bundles, and so on. When described abstractly, many results of field theory are also incorporated, including classical algebro-geometric problems related to finding algebraic points on varieties.
