1/ To say that a topological set is "complete" has a mathematical meaning. For example, one definition of completeness is that all Cauchy-sequences converge.
A number is not complete (or at least I have never heard of such a definition for numbers).
We say that the real numbers are complete, because they satisfy the definition.
2/ When we have a sequence, we can (sometimes) see if this sequence converge or not. Again, the notion of convergence has a mathematical meaning, and suppose that the sequence obey a certain definition. When a sequence converges, we name an element "limit", this is an element that obey a certain definition too. It is straightforward from the definitions of convergence and of limit, that if a limit exists, it is unique.
For example we can study the sequence given by 0, 0.9, 0.99, 0.999, ... This sequence converges and has limit 1.
Or we can study another sequence $\frac{p_n}{q_n}$ given by the conditions $q_n = 10^n$ and $\left(\frac{p_n}{q_n}\right)^2 \leq 2 \leq \left(\frac{p_n+1}{q_n}\right)^2$. This sequence will also converge, and the limit will be $\sqrt{2}$.
Now, if observe that this last example is a sequence composed only by elements in $\mathbb{Q}$, we might be surprised to see that the limit is actually not in $\mathbb{Q}$. Mathematicians don't like the fact of not having the limit within the working topological set. Therefore they build $\mathbb{R}$ which is the set all (classes of equivalence of) Cauchy-sequences of elements in $\mathbb{Q}$. Hence $\mathbb{R}$ satisfy the definition of completeness.
Again, saying that $\mathbb{R}$ is complete is just a matter of definition. It has nothing to do with the word "gap". As far as I know, "gap" has no mathematical meaning.