My own knowledge of algebraic geometry is not yet even at a "cocktail party" level, but I'd also love to learn a bit about why (and how) I should care about AG. But I have two points to mention here.

1. Lior Pachter and Bernd Sturmfels have edited a book called Algebraic Statistics for Computational Biology, and in that book they argue how the language of algebraic geometry offers some help for tackling problems in computational biology + statistics.

2. Another exciting perspective might be offered by the work of Ketan Mulmuley on Geometric complexity theory, where Mulmuley is using algebraic geometric ideas to tackle the P vs NP problem.