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(Everything I say here is up to homotopy equivalence.)

Algebras over $\Omega^k \Sigma^k$ are spaces $X$ equivalent to a $k$-fold loop space $\Omega^k Y$. Algebras over $\Omega^\infty \Sigma^\infty$ are infinite loop spaces; this is a little harder to say, but it is essentially that there is a sequence of spaces $Y_n$ with $Y_0 = X$ and equivalences $Y_n \simeq \Omega Y_{n+1}$.

The original and still canonical reference, which covers all of this in detail, is J.P. May's book "The geometry of iterated loop spaces," Lectures Notes in Mathematics 271.

EDIT: As Neil points out, I've misread the question. The statements above are for spaces, not for objects in the homotopy category of spaces.

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(Everything I say here is up to homotopy equivalence.)

Algebras over $\Omega^k \Sigma^k$ are spaces $X$ equivalent to a $k$-fold loop space $\Omega^k Y$. Algebras over $\Omega^\infty \Sigma^\infty$ are infinite loop spaces; this is a little harder to say, but it is essentially that there is a sequence of spaces $Y_n$ with $Y_0 = X$ and equivalences $Y_n \simeq \Omega Y_{n+1}$.

The original and still canonical reference, which covers all of this in detail, is J.P. May's book "The geometry of iterated loop spaces," Lectures Notes in Mathematics 271.