In infinite dimensional spaces there may be two different types of "bases".
A "Hamel basis" is an infinite set such that any vector in the vector space can be written as a linear combination of a finite number of vectors in the basis. The functions 1, x, x^2, ...,x^n, ... form a Hamel basis for the space of all polynomials but not for the set of (real) analytic functions (a real valued function on the real numbers is "analytic" if and only if it is equal to its Taylor series). It can be shown (assuming axiom of choice) that every vector space has a Hamel basis.
If you have a topology on your vector space, and so a notion of "convergence", a more general concept of "basis" is, as LKurtz said, a set of vectors such that any vector can be written as a possibly infinite linear combination. The functions 1, x, x^2, ...,x^n, ... form a basis, in this sense, for the space of all (real) analytic functions.