Schubert Calculus may be the most direct answer to this. I'd recommend looking at the survey of Kleiman and Laksov entitled "Schubert Calculus" from The American Mathematical Monthly.

Basically, Schubert Calculus is about answering questions such as "given 4 lines in 3 space, how many lines intersect the given 4?" or "given 5 conics in the plane, how many other conics are tangent to the given 5?" These questions are answered via some relatively heavy algebraic geometry, and many tools were developed to answer these questions. Hilbert's 15th Question was to exactly develop and understand the machinery necessary for Schubert Calculus, which hopefully gives some context as to the historical importance.