Broadly speaking, algebraic geometry is the geometric study of solutions to polynomial equations. To begin with, you would start by working with solutions in affine space $\mathbb{A}_k^n= k^n$, where $k$ is an algebraically closed field (e.g. $\mathbb{C})$. Eventually, it becomes advantageous to add points at infinity by working in projective space $\mathbb{P}^n_k=\mathbb{A}^n_k\cup (\text{hyperplane at }\infty)$. After a while, you may move beyond even this.
All of this is spelled out in the basic books on algebraic geometry. Although there are quite a number of these, very few are explicitly at the undergraduate level. If I had to come up with a list, it would probably be the similar to Davidac897's. Out of the lot, I'd recommend Reid's "Undergraduate Algebraic Geometry" as a good starting point: it is short, to the point, with minimal prerequisites. Once you've gotten through it, you'll have enough of a foundation to tackle something more ambitious if you're still so inclined.
One other thought. I have a sense that you are planning to read this on your own. Things are easier if you find a group of fellow students to "share the pain". Have fun.
