If $G$ has $|G|$ many elements, then the set
$$\{a, a^2, a^3, \cdots, a^{|G|}\}$$
either has a repetition or exhaust the group. If it exhaust the group, one of these elements is $1$. If there is a repetition, so $a^{r} = a^{s}$ for $r>s$, then $a^{r-s} = 1$. In any event, at least one of these elements must equal $1$, say $a^n = 1$, and choose $n$ to be the smallest such exponent. If we can show that $n$ divides $|G|$, we are done. But $n$ is the order of the subgroup
$$\{a, a^2, \cdots , a^n(=1)\}$$
And by Lagrange's theorem, the order of a subgroup must divide the order of the group.
To rephrase here a bit: If $a$ is a generator of $G$, then raising it to the power of the order of the group guarantees that it will cycle through all the elements and return to the identity. If $a$ is not a generator of $G$ then Lagrange's theorem guarantees that the order of the subgroup generated by $a$ divides $|G|$ and therefore if $a^{|a|}=1$ then $(a^{|a|})^{u}=1$ where $|G|=u|a|$.
In response to your questions about definitions: A group is an abstract object. We do not know anything about the sorts of objects inside the group. All we know about the group is that it satisfies certain axioms and has a (binary) group operation $\ast$. Given a pair of elements $a,b \in G$, we write $a \ast b$ or $ab$ to denote the group operation acting on the pair of elements. The exponent notation means: $a^{2} = aa, a^{7} = aaaaaaa$.
Best Answer
Generally if $G$ is a group with the neutral element $e\in G$, $g\in G$ is any element and $n\in\mathbb{Z}$ is an integer then $g^n$ is defined by the following rules:
$$g^0:=e$$ $$g^n:=gg^{n-1}\text{ for }n>0$$ $$g^n:=(g^{-1})^{-n}\text{ when }n<0$$
The middle rule is recursive. In the last one note that when $n<0$ then $-n>0$ and so $(g^{-1})^{-n}$ is well defined by the previous rule.
Few examples that follow from this definition:
$$g^1=gg^0=ge=g$$ $$g^2=gg^1=gg$$ $$g^3=gg^2=ggg$$ $$g^{-2}=(g^{-1})^2=g^{-1}g^{-1}$$ $$g^{-3}=(g^{-1})^3=g^{-1}g^{-1}g^{-1}$$
and so on.