Generally speaking, studying what the solution sets of polynomial equations is "like" is quite fundamental to a lot of mathematics. Doing this in a "deep" way can lead to a reimagining of much of modern mathematics: https://rawgit.com/iblech/internal-methods/master/notes.pdf

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For instance, lots of people use a straightforward generalisation of number systems called rings. But ring theory is quite abstract. Modern Algebraic Geometry shows that at least in the case of commutative rings, these are merely spaces of functions on a space called a ring's spectrum. You can visualise a ring's spectrum, unlike the ring itself. Many properties of a ring are just properties of its spectrum. This seems like a significant conceptual leap in the understanding of things that were studied since the 1800s without much geometric understanding.

Oh yeah, and I'm not an algebraic geometer.

- consolidation of many types of results into a `simple' theoretical framework, I suppose this originates with Noether, and reaches it apotheosis in Bourbaki's tracts.

- embedding of 'classical' objects (solutions to polynomial equations) inside a larger 'category' (schemes) where certain mysterious relations observed in the classical world (Weil Conjectures) have a more `natural' interpretation (fixed point theorem) and light the way to a proof which would have otherwise been beyond reach

y^2 - x^3 - x = 0, x^2+y^2+1=0

But actually you do not need to talk about the underlying space directly. If you want to talk about a space, all you actually need to think about are the possible functions on the space. If you want to talk about geometry, you only need the algebra of functions on that space, so in the example just the polynomials themselves, rather than having to say explicitly solve it for the points. You can use this big idea in a lot of other areas of mathematics and physics.