Spanier-Whitehead stabilization provides a way to extend a category $\bf E$ to a bigger one $\mathcal{SW}\_\Omega(\bf E)$ where a given endofunctor $\Omega$ is invertible. The category $\mathcal{SW}\_\Omega(\bf E)$ is constructed with


1. Objects the pairs $(A,n)\in Ob(\mathbf C)\times\mathbb Z$;
2. The set of morphisms $(A,n)\to (B,m)$ corresponds to the colimit set
$$
 \varinjlim\_{k\in\mathbb N} \hom\_{\bf E}(\Omega^{n+k}A, \Omega^{m+k}B)
$$


It's a matter of bare computations to show that it defines a category, where $\bf E$ can be embedded via $A\mapsto (A,0)$, and where an entire family of functors $\bar\Omega^i\colon (A,n)\mapsto (A,n+1)$ can be defined; the functor $\bar\Omega^1$ plays the role of the initial endofunctor $\Omega$, and that's the end of the story.


But when you meet the formalism of stable $\infty$-categories you begin to wonder if there's a link between the two processes, the SW stabilization and what Lurie describes [here](http://arxiv.org/pdf/math/0608228v5.pdf) (Def. 8.4). I'm aware that the SW construction is an "abstraction" (?) of the procedure exhibiting topological spectra, but I must confess I'm not able to go further (especially because I'm a "category theorist" slightly oriented to topology, not vice-versa).



Stabilization of $\infty$-categories versus SW stabilization
I think it's more typical to use this construction to invert the suspension functor, rather than the loop functor. So let me write $\Sigma$ where you wrote $\Omega$.


1) The category $SW\_{\Sigma}(\mathcal{C})$ can be identified with the direct limit of the sequence
$$ \cdots \rightarrow \mathcal{C} \stackrel{\Sigma}{\rightarrow} \mathcal{C} \stackrel{\Sigma}{\rightarrow} \mathcal{C} \stackrel{\Sigma}{\rightarrow} \cdots$$


2) The construction makes sense for $\infty$-categories as well as ordinary categories. Moreover, it commutes with passage to homotopy categories. That is, if $\mathcal{C}$ is an $\infty$-category, then the homotopy category of $SW\_{\Sigma}(\mathcal{C})$ is $SW\_{\Sigma}( h \mathcal{C} )$. 


3) Let $\mathcal{C}$ be a small pointed $\infty$-category which admits finite colimits, and let $\Sigma: \mathcal{C} \rightarrow \mathcal{C}$ be the suspension functor. Then $Ind( SW\_{\Sigma}( \mathcal{C} ) )$ is the stabilization of $Ind( \mathcal{C} )$. 


You'd typically put these together by taking $\mathcal{C}$ to be something like the $\infty$-category of pointed finite spaces. Then $Ind( \mathcal{C} )$ is the $\infty$-category of pointed spaces, and its stabilization is the $\infty$-category of spectra. Your conclusion is that $SW\_{\Sigma}(\mathcal{C} )$ can be identified with the $\infty$-category of finite spectra, so that
$SW\_{\Sigma}( h \mathcal{C} )$ is a model for the homotopy category of finite spectra (this is the classical construction).


