Unary encoded language set
Forget about strings meaning anything. For any symbol&nbsp;$a$ and natural number&nbsp;$n$, $a^n$&nbsp;is the string made from $n$&nbsp;copies of&nbsp;$a$. So $a^1=a$, $a^2=aa$, $a^5=aaaaa$ and $a^0$&nbsp;is the empty string, usually written as $\varepsilon$ or&nbsp;$\lambda$.

The set $\{1^n\mid n\in\mathbb{Z}\}$ isn't well-defined (did Sipser really write that?), since $-1\in\mathbb{Z}$ but $1^{-1}$ doesn't make sense: what would $-1$ copies of the symbol&nbsp;$1$ look like?

However, the set $\{1^n\mid n\in\mathbb{N}\}$ is just $\{1^0, 1^1, 1^2, 1^3, \dots\}$, i.e., $\{\varepsilon, 1, 11, 111, \dots\}$ (I'm considering zero to be a natural number; delete $\varepsilon$ if you think that one is the first natural number). Now, strings such as $111$ can be interpreted as numbers written out in some base (unary, binary, decimal, whatever). But the definition is just giving you a set of strings.