Perhaps here is some part of the story:
- Algebraic curves = 1-dimensional algebraic varieties (shapes cut out by polynomial equations) - in some sense they're the simplest (but non-trivial!) shapes you can study. (1-dimensional objects is easier than higher dimensional objects! )
- In studying geometry, one common thing to look at is the collection of functions (satisfying some properties) that you can define on your object. For algebraic varieties, the functions you care about are functions given by polynomials. If you're working over \C, you can think of holomorphic functions instead.