The Zariski topology is part of the basic structure of varieties and schemes. Unlike other, fancier Grothendieck topologies, it is actually a topology, defined by subsets of the variety/scheme, and so gives rise to the notion of closed, as well as of open, subset. The closed subsets are the algebraic subsets (in the variety case) or (the spaces underlying) the closed subschemes (in the scheme case), which are what the study of algebraic geometry is to a large extent about.
If you look at Hartshorne, Chapters IV and V, you will find a lot of geometry of curves and surfaces (as well as some geometry of higher dimensional varieties too), all of it treated without recourse to any topology other than the Zariski topology.
Added: A good example, in my opinion, is the (direct) proof that the addition law on a smooth plane cubic curve is associative. (By direct, I mean the proof working from the definition of the group law in terms of collinear points, not the more involved proof that proceeds by identifying the curve with its Picard variety.) It is not hard to see that associativity holds when the three points being added are in suitably general position. To conclude, one can either make complicated special cases arguments in the various non-generic situations, or, one can make a continuity argument in the Zariski topology. The latter argument is simple (in that it uses the most basic kinds of arguments about continuous maps in general topology) and decisive. It illustrates perfectly the role of the Zariski topology in geometric arguments.
One could generalize from this example as follows: two of the most fundamental notions of algebraic geometry are the complementary concepts of general and special position, and these notions are precisely what the Zariski topology captures (just as the topology on a metric space captures the notion of closeness in the metric).