As a novice in algebraic topology, I'm trying to grasp the concept of a spectrum. Let me first sketch two motivations.

One motivation goes like this: for singular cohomology of spaces, we have $H^n(X;G)=[X,K(G,n)]$. We observe that these Eilenberg-Mac Lane spaces are such that the maps $K(G,n)\to \Omega K(G,n+1)$ are weak homotopy equivalences. We get the notion of an $\Omega$-spectrum, and realize that any such spectrum gives rise to an (extraordinary reduced) cohomology theory in this way. We can then prove that actually any such cohomology theory on CW complexes arises in this fashion. Thus a spectrum is a cohomology theory (well, an $\Omega$-spectrum actually, but never the matter).

Here's another one. Spectra, whatever that will be, should be the objects of a category where stable phenomena occur. So for instance, Freudenthal's theorem says that for $X$ a finite CW complex, the suspension colimit $colim_k [\Sigma^k X,\Sigma^k Y]$ stabilizes for sufficiently large $k$. We would like this colimit to be a the group of morphisms from a spectrum associated to $X$ to a spectrum associated to $Y$. This will be what the suspension spectra will accomplish: that colimit will be exactly $[\Sigma^\infty X,\Sigma^\infty Y]$.

This is all fine and well, but I don't understand how *the same object* can come to realize both objectives at the same time. Moreover, both point of views are used at the same time when, for instance, we consider $E^*(E)$ for a spectrum $E$ (which is what we can quickly identify with the group of stable $E$-cohomological operations).

Is there some deep reason behind this? Maybe both motivations are actually the same, if we look at them at a higher level?

stable homotopy categoryis something that allows us to study both these phenomena at once (as I mentioned above). There are many different definitions of the stable homotopy category, for example, using $\mathbf{S}$-modules, symmetric spectra, orthogonal spectra, etc. Stable homotopy theory is the study of the stable homotopy category. (See this as well: math.stanford.edu/~carym/stable.pdf) $\endgroup$