In a sentence, algebraic geometry is the study of solutions to algebraic equations. People learning it for the first time, would see a lot of algebra, but not much geometry. But it is there. The picture above depicts a resolution of the singular curve y

For a really elementary introduction, go here.

For a more serious introduction, you can get my notes on basic algebraic geometry. This is sort of a "prequel" to Hartshorne. (It's a 340K PDF file)

My notes "Algebraic geometry over the complex numbers" covers more: sheaf theory, cohomology and Hodge theory. (It's a 1.3 MB PDF file).