affine space
Affine space is the blank canvas of algebraic geometry: just tuples of coordinates, with no distinguished origin singled out as special and no inner product imposed. It is the ambient stage on which affine varieties are drawn, the algebraic-geometry counterpart of plain coordinate space.
As a point set, affine n-space A^n over a field k is the set k^n of n-tuples (a_1, ..., a_n). What makes it geometric is the extra structure: it carries the Zariski topology, and its ring of polynomial functions is k[x_1, ..., x_n]. When k is not algebraically closed one usually still calls A^n the functor whose points over an extension K are K^n, so that A^1 over Q already 'knows' about the algebraic numbers.
A subtle point distinguishes affine space from a vector space. As an abstract algebraic variety A^n has no preferred zero vector and no linear structure baked in; one is free to translate the origin anywhere. In the scheme language A^n becomes Spec of the polynomial ring, whose points include not just the classical tuples but also generic points corresponding to lower primes.
A^1 over an algebraically closed field is the affine line; its closed sets are the finite subsets together with the whole line, so it is far from Hausdorff.
Even the simplest affine space already shows the unusual coarseness of the Zariski topology.