The Archimedean Property of $\mathbb{R}$ comes into two visually different, but mathematically equivalent versions:
Version 1: $\mathbb{N}$ is not bounded above in $\mathbb{R}$.
This essentialy means that there are no infinite elements in the real line.
Version 2: $\forall \epsilon>0\ \exists n\in \mathbb{N}:\frac1n<\epsilon$ This essentially means that there are no infinitesimally small elements in the real line, no matter how small $\epsilon$ gets we will always be able to find an even smaller positive real number of the form $\frac1n$.
Note that $0$ is not infinitesimally small as it is not positive (remember that we take $\epsilon>0$) and $\infty$ doesn't belong in the real line. The extended real line $\overline{\mathbb{R}}$ is in fact not Archimedean, not only because it has infinite elements, but because it is not a field! ($+\infty$ has no inverse element for example).
You may want to note that the Archimedean Property of $\mathbb{R}$ is one of the most important consequences of its completeness (Least Upper Bound Property). In particular, it is essential in proving that $a_n=\frac1n$ converges to $0$, an elementary but fundumental fact.
The notion of Archimedean property can easily be generalised to ordered fields, hence the name Archimedean Fields.
Now, surreal numbers are not exactly $\pm \infty$ and I suggest you read this Wikipedia entry. You might also want to read the Wikipedia page for Non-standard Analysis. In non standard analysis, a field extension $\mathbb{R}^*$ is defined with infinitesimal elements! (of course that's a non Archimedean Field but interesting enough to study)