As homotopy theory is the study of homotopy types, so stable homotopy theory is the study of stable homotopy types. As homotopy theory in generality is (∞,1)-category theory (or maybe (∞,1)-topos theory), so stable homotopy theory in generality is the theory of stable (∞,1)-categories.
More specifically, if one thinks of classical homotopy theory as the study of (just) the (∞,1)-category $L_{whe}$Top $\simeq$ ∞Grpd of topological spaces modulo weak homotopy equivalence (∞-groupoids), or rather of its homotopy category $Ho(Top)$, then stable homotopy theory is the study of the corresponding stabilization: To every suitable (∞,1)-category is associated its corresponding stable (∞,1)-category of spectrum objects. For $L_{whe}$Top this is the stable (∞,1)-category of spectra, $Sp(L_{whe}Top)$. Stable homotopy theory is the study of $Sp(Top)$, or rather of its homotopy category, the stable homotopy category $Ho(Sp(L_{whe}Top))$.
By definition a stable homotopy type is one on which suspension and hence looping and delooping act as an equivalence. Historically people considered in plain homotopy theory statements that became true after sufficiently many suspensions, hence once the process of taking suspensions “stabilizes”. Whence the name.
The study of monoid object in a monoidal (infinity,1)-category in a stable (∞,1)-category is the homotopy-theoretic version of commutative algebra, hence higher algebra and higher linear algebra.
A tool of central importance in stable homotopy theory and its application to higher algebra is the symmetric monoidal smash product of spectra which allows us to describe A-∞ rings and E-∞ rings as ordinary monoid objects in a model category that presents $Sp(Top)$. (“brave new algebra”).
When the spaces and spectra in question carry an infinity-action of a group $G$ the theory refines to
A quick idea is given in
A fun scan of the (pre-)history of the stable homotopy category is in
An excellent general survey of modern stable homotopy theory is in
A survey of formalisms used in stable homotopy theory – tools to present the triangulated homotopy category of a stable (infinity,1)-category – is in
Neil Strickland, Axiomatic stable homotopy - a survey (arXiv:math.AT/0307143)
Mark Hovey, John Palmieri, Neil Strickland, Axiomatic stable homotopy theory (pdf)
Stefan Schwede, Symmetric spectra (pdf)
An account in terms of (∞,1)-category theory is in section 1 of
Brief indications of open questions and future directions (as of 2013) of algebraic topology and stable homotopy theory are in