symmetric monoidal (∞,1)-category of spectra
A ring spectrum is a model for an A-∞ ring.
The notion of spectrum is the generalization of the notion of abelian group to homotopy theory/(∞,1)-category theory. The notion of ring spectrum is the corresponding generalization of the notion of ring.
A ring spectrum is a monoid in the stable homotopy category $Ho(\mathcal{S})$ equipped with the smash product of spectra. See there for more details.
This means that a ring spectrum is a monoid in the category $\mathcal{S}$ of spectra up to not-necessarily coherent homotopy.
A monoid-up-to-homotopy in the category of spectra for which the homotopies are coherent is called an $A_\infty$-ring spectrum? or just an $A_\infty$-ring. These may be modeled as monoids with respect to the symmetric monoidal smash product of spectra.
Abstractly these are E1-algebras in the symmetric monoidal (∞,1)-category Spec of spectra.
Not every ring spectrum may be refined to an $A_\infty$-ring spectrum.
An account in terms of (∞,1)-category theory is in section 7.1 of
An account in terms of model categories is in
See also the references at stable homotopy theory.