I am learning about sequences and for example:
A series $\sum_{k = m}^ns_k$ is convergent iff for every $\epsilon > 0[\exists N : m, n > N \Longrightarrow |\sum_{k = m}^ns_k| < \epsilon]$.
Now in this example, if it wasn't for the Archimedan property then this wouldn't hold true - there would be an infinitesimal/infinite element such that any sum is greater than said infinite(simal) element.
In fact, if it wasn't for the property, most of real analysis would fail - the definition of $\epsilon$ involves $\exists n : \frac{1}{n} < \epsilon$.
Does this also have to do with the density of $\mathbb{Q}$ in $\Bbb{R}$? Topologically perhaps?
I would say "no". $\mathbb Q$ is Archimedean, but would be terrible for analysis. In contrast, things like the Levi-Civita field are non-Archimedean, but their non-Archimedean analysis is still studied.
What does have to do with the density of $\mathbb Q$ in $\mathbb R$ is the fact that real epsilons are unnecessary: As you point out, for all positive $\varepsilon$, you can find a positive rational number smaller than it. That flavor of statement doesn't really require the Archimedean property, though, as it's also true for instances of the Hyperreals.
As Brian M. Scott said and linked, the least upper bound property leads to the Archimedean property almost immediately, and less obviously, it leads to the field being isomorphic to the reals, which you can read a proof of in a textbook like Spivak's Calculus or in the undergrad thesis Pete L. Clark linked. Since you asked about "real analysis", and the least upper bound property is one of many equivalent defining properties of the reals (taken as given that it's an ordered field, which is certainly not always the case in analysis), I'd have to agree with Prahlad Vaidyanathan that that could be the "most important" property.