I'm trying to understand how a given presentation of a group is well defined. It says on Wikipedia (https://en.wikipedia.org/wiki/Presentation_of_a_group) that $G \ =\ \langle a\ |\ a^n=1\rangle$ is a presentation for the cyclic group of order $n$. But to me, that presentation isn't well defined. We're not given that the order of $a$ is $n$, just that $a^n=1$. If $n=8$ say then the trivial group, the cyclic group of order $2$ and the cyclic group of order $4$ also satisfy the conditions of the presentation. How is it that $G$ is actually uniquely defined?
Also, I don't know anything about free groups, quotient groups, normal subgroups or any of that stuff that Wikipedia uses to define group presentations. I'm self-studying Dummit and Foote and they introduce group presentations early on without any of those concepts.
Roughly speaking, the main idea is that the group defined by a presentation is the "largest" or "freest" group that satisfies the presentation.
In particular, a group satisfying the presentation does not mean that it is the presented group. (It only means that it is a quotient of the presented group.)
For example, in your case, $G$ should be the "largest" group on a generator called $a$ such that $a^n=1$, not just any group where $a^n=1$ is satisfied, and it turns out that $G$ is the cyclic group of order $n$.
To make this more formal (including the fact that this really defines a unique group), you need a to learn a bit about free groups and so on.
See also: Is a group defined by its generator set and relations?
(In fact, one could argue this is a duplicate.)