Banach space
A Banach space is a normed vector space with no holes. The norm gives you a way to measure distances, and completeness guarantees that any sequence whose terms crowd arbitrarily close together actually has a limit living inside the space. This is the same upgrade that turns the rationals (full of gaps) into the real numbers, only now performed on spaces of functions, sequences, or operators.
Formally, a Banach space is a normed vector space (V, ||·||) that is complete as a metric space under d(x, y) = ||x - y||: every Cauchy sequence in V converges to a point of V. Completeness is precisely what lets you construct objects as limits — solutions of differential equations, fixed points of contractions, sums of infinite series of functions — and be sure the limit exists where you expect it.
Most of the spaces analysts care about are Banach spaces: the continuous functions C[a, b] under the sup norm, the Lebesgue spaces L^p for 1 <= p <= infinity, the sequence spaces l^p. A normed space that fails to be complete can always be enlarged to its completion, a Banach space in which the original sits as a dense subset. The polynomials under the sup norm on [0, 1] are not complete; their completion is exactly C[0, 1] by the Weierstrass approximation theorem.
The space C[0, 1] with the sup norm is complete: if f_n is uniformly Cauchy then f_n converges uniformly to some f, and the uniform limit of continuous functions is continuous, so f lies in C[0, 1]. By contrast the same functions under the L^1 norm integral|f| are NOT complete — Cauchy sequences can converge to a discontinuous function outside the space.
Same vector space, two norms: complete under the sup norm, incomplete under the L^1 norm.
Completeness is invisible to the algebra and even to the topology of bounded sets — it is a property of limits. The deepest theorems of the subject (open mapping, closed graph, uniform boundedness) all secretly run on the Baire category theorem, which requires completeness and fails without it.