I'm not going to add nothing directly related to your question and previous answers, but make some propaganda of a theorem I like since I was student and which, I believe, says something stronger than comparing some intuitive notion of completness with its definition.
A somewhat related notion of completeness is the geodesical one. The definition may not be too much appealing unless you're interested in differential geometry, but one of its consequences is easy to explain: if a Riemann manifold is geodesically complete, you can join any two points by a length minimizing geodesic. (But geodesic already implies that it minimizes length, doesn't it? Not quite: just locally. So, for instance, the meridian joining the North Pole with London, but going "backward", through the Bering Strait and the Pacific Ocean, then the South Pole, Africa and finally London, is a geodesic, but not a length minimizing one blatantly.)
Anyway, $\mathbb{R^2} \backslash \left\{ (0,0)\right\} $ is not geodesically complete, since there is no length minimizing geodesic joining, say, $(-1,0)$ and $(1,0)$, due to the "hole" $(0,0)$. At the same time, as a metric space, $\mathbb{R^2} \backslash \left\{ (0,0)\right\}$ is not complete: the Cauchy sequence $(\frac{1}{n}, 0)$ converges to $(0,0)$, but since $(0,0)$ is not in $\mathbb{R^2} \backslash \left\{ (0,0)\right\}$ it doesn't have a limit there.
Well, the Hopf-Rinow theorem tells us that this kind of things always happen together: a "hole" for geodesics is the same as a "hole" for Cauchy sequences, since for a (finite-dimensional) Riemann manifold $M$, both notions agree: $M$ is complete as a metric space if and only if it is geodesically complete.
The term "localization" is indeed taken from geometry. Consider the ring $R = C(\mathbb{R})$ (continuous real valued function on the real numbers) and the set $U$ of function that don't vanish at the origin. Clearly $U$ is multiplicative. Now, what is the geometric interpretation of $U^{-1}R$ ? If we are only interested in the behavior of functions near the origin, a function that doesn't vanish at the origin doesn't vanish in some open neighborhood of the origin. Therefore, by restricting such function to a small enough neighborhood, we get a non-vanishing function and therefore it is possible to take its (multiplicative) inverse. Thus, $U^{-1}R$ can be thought of as the result of concentrating attention to small neighborhoods of the origin and hence the name "localization".
In a general commutative ring we adopt the intuition of the geometric case and think of elements of the ring as "functions" on some "geometric object" whose points are the prime ideals of the ring. The full picture of this is given by the modern algebro-geometric concept of a scheme.
Best Answer
Ideals were originally defined in analogy to numbers; in fact, "ideal" was used in place of Kummer's "ideal numbers" (which were introduced to provide a kind of "unique factorization" in the ring of cyclotomic integers $\mathbb{Z}[\zeta_p]$).
In this setting, the general philosophy is to translate divisiblity statements about numbers into statements about ideals, since every number corresponds to an ideal (the principal ideal it generates), but there may be ideals that don't correspond to "actual numbers" (the non-principal ideals).
In analogy to the integers, a number $p$ is prime if and only if $p\neq\pm 1$ and if $p|ab$, then $p|a$ or $p|b$. In the ideal-theoretic setting, divisibility was equivalent to containment, so the condition would be translated to "$(p)\neq (1)$ and $(ab)\subseteq (p)$ implies $(a)\subseteq (p)$ or $(b)\subseteq (p)$." Moving from principal to general ideals, we say the ideal $P$ is prime if and only if $P\neq R$ and if $AB\subseteq P$, then either $A\subseteq P$ or $B\subseteq P$.
From here, once the notion of ring was generalized away from rings-of-integers of number fields, the notion was kept.
(For commutative rings, the definition is equivalent to the statement "if $ab\in P$, then $a\in P$ or $b\in P$ ", but for noncommutative rings this condition is stronger; that is, if an ideal satisfies the element-theoretic version, then it is prime; but it can be prime and not satisfy the element-theoretic version; for example, in the ring of $2\times 2$ matrices over $\mathbb{R}$, the trivial ideal $(0)$ is prime, but there are certainly pairs of matrices, neither of them the zero matrix, whose product is the zero matrix.)
Addendum. Here is how Dedekind put it in Sur la Théorie des Nombres Entiers Algébriques (1877), translated as Theory of Algebraic Integers by John Stillwell, Cambridge University Press, 1966:
Later, Dedekind proves that in this context, $\mathfrak{m}\subseteq \mathfrak{n}$ if and only if there exists $\mathfrak{r}$ such that $\mathfrak{n}\mathfrak{r}=\mathfrak{m}$, establishing the link between "divisibility" in terms of inclusion, and divisibility in terms of multiplication, which holds in these kinds of rings but not in general. He called it the hardest part of the development.