In algebraic geometry, the Zariski topology is a particular topology chosen for algebraic varieties that reflects the algebraic nature of their definition. It is due to Oscar Zariski and took a place of particular importance in the field around 1950. The more subtle étale topology is a refinement of the Zariski topology discovered by Grothendieck in the 1960s that reflects the geometry more accurately.
Read more about Zariski Topology: The Classical Definition, The Modern Definition