Zariski topology
Ordinary topology measures nearness with little balls. The Zariski topology measures nearness with polynomials: a closed set is one carved out by equations, and an open set is the complement, where at least one polynomial is allowed to be nonzero. The result is a very coarse topology, perfectly tuned to algebra rather than to analysis.
On affine n-space the closed sets are exactly the vanishing sets V(I) of ideals I; one checks these are closed under arbitrary intersection and finite union, so they form the closed sets of a topology. On the prime spectrum Spec R of a commutative ring the same recipe works: closed sets are V(I) = { primes p containing I }, giving a topology on the set of prime ideals. The two pictures agree on classical points and extend the notion to all rings.
The Zariski topology is strange by analytic standards. It is almost never Hausdorff — on the affine line any two nonempty open sets meet, since they omit only finitely many points. Irreducible varieties are precisely the spaces that cannot be split into two proper closed pieces, and a nonempty open subset of an irreducible space is dense and again irreducible. These features are not defects; they encode the rigidity that makes algebraic geometry work.
On A^1 over C the proper closed sets are the finite point sets, so the open sets are the cofinite sets together with the empty set; this makes A^1 irreducible.
Cofinite opens: any two nonempty opens overlap, the hallmark of irreducibility.
A basis for the Zariski topology is given by the distinguished open sets D(f) = { points where f is nonzero }; on Spec R the ring of regular functions on D(f) is the localization R[1/f].