symmetric monoidal (∞,1)-category of spectra
Given a symmetric monoidal (infinity,1)-category $C$, an $E_\infty$-algebra in $C$ is synonymous with a commutative monoid in $C$. Most often $C$ is the (infinity,1)-category of chain complexes over a field or commutative ring, or the stable (infinity,1)-category of spectra (see also E-infinity ring).
In terms of operads, an $E_\infty$-algebra is an algebra over an operad for an E-∞ operad.
$E_\infty$-algebras in chain complexes are equivalent to those in abelian simplicial groups.
For details on this statement see monoidal Dold-Kan correspondence and operadic Dold-Kan correspondence.
The singular cohomology $H^*(X,\mathbb{Z})$ of a topological space is a graded-commutative algebra over the integers, but the the singular cochain complex $C^*(X,\mathbb{Z})$ is not: instead, it is an $E_\infty$-algebra. Remarkably, for simply connected spaces of finite type this $E_\infty$-algebra knows everything about the weak homotopy type of $X$. In fact a stronger statement holds:
Finite type nilpotent spaces $X$ and $Y$ are weakly homotopy equivalent if and only if the $E_\infty$-algebras $C^*(X,\mathbb{Z})$ and $C^*(Y,\mathbb{Z})$ are quasi-isomorphic.
This was proved by Mandell in 2003.
A connected space of the homotopy type of a CW-complex with a non-degenerate basepoint that has the homotopy type of a $k$-fold loop space for all $k \in \mathbb{N}$ admits the structure of an $E_\infty$-space.
The model structure on algebras over an operad over E-∞ operads in Top and in sSet are Quillen equivalent.
This is in BergerMoerdijk I, BergerMoerdijk II.
An $E_\infty$-algebra in spectra is an E-∞ ring.
See Ek-Algebras.
See symmetric monoidal (∞,n)-category.
In the context of (infinity,1)-operads $E_\infty$-algebras are discussed in
A systematic study of model category structures on operads and their algebras is in
The induced model structures and their properties on algebras over operads are discussed in