I know very little about algebraic topology, and more about $k$-linear stable $\infty$-categories (i.e. homological algebra).

Given an abelian group $A$, there is the Eilenberg-Mac Lane spectrum $HA$, for which: $\pi_{< 0}(\cdot) = 0$, $\pi_0 (\cdot) = A$, $\pi_{>0}(\cdot) = 0$. There is also the Moore spectrum $SA$, for which $\pi_{<0}(\cdot) = 0$, $\pi_{=0} (\cdot)= A$, $H_{> 0} (\cdot , \mathbb{Z}) = 0$.

I more-or-less have some feeling for $HA$ (under Dold-Kan, it is simply $A$, so one simply gets connected deloopings $BA$, $BBA$, etc. making the infinite loop space structure explicit).

I have no feeling what-so-ever about $SA$.

But reading about Bousfield localization, it seems that most the standard examples are with $SA$. How to understand this for a novice? Say, for $p$-localization and $p$-completion, how do I understand that I "want" to use the Moore spectrum and not the Eilenberg-Mac Lane spectrum?

Or, at least, what is the rationale for introducing Moore spectra?

Thank you

coulduse HZ/p, and here the answer is somewhat unintuitive: for connective spectra, localization at S/p and at HZ/p agree, and so you cannot tell the difference before passing to nonconnective spectra, which are somewhat exotic objects. The spectrum $KU := \Sigma^\infty_+ \mathbb CP^\infty[\beta^{-1}]$ is a concrete example of a spectrum with $H_*(KU; Z)$ rational, hence $H_*(KU; Z/p)$ and $L_{HZ/p} KU$ null, but $L_{S/p} KU$ nontrivial. $\endgroup$