Among sequential pre-spectra $X$, the $\Omega$-spectra are those for which the structure map $X_n \to \Omega X_{n+1}$ is a weak homotopy equivalence (beware that some authors require a homeomorphism instead and say “weak $\Omega$-spectrum”, for the more general case).
Omega-spectra are particularly good representatives among pre-spectra of the objects of the stable (∞,1)-category of spectra, hence of the stable homotopy category. For instance they are (after geometric realization) the fibrant objects of the Bousfield-Friedlander model structure.
With $\Omega$ the notation for the loop space construction (whence the name),an $\Omega$-spectrum is a sequence $E_\bullet = (E_n)_{n \in \mathbb{N}}$ of pointed ∞-groupoids (homotopy types) equipped for each $n \in \mathbb{N}$ with an equivalence of ∞-groupoids
Remark: In terms of model category presentation one may also consider sequences of topological spaces equipped with homeomorphisms $E_n \longrightarrow \Omega E_{n+1}$ See at spectrum the section Omega-spectra.
The inclusion of $\Omega$-spectra into all sequential pre-spectra has a left adjoint, spectrification. See there for more.
Given a pointed topological space $X$, its suspension spectrum $\Sigma^\infty X$ is far from being an $\Omega$-spectrum. The $\Omega$-spectrum that it induces (its spectrification) is given by free infinite loop space constructions:
write
for the free infinite loop space functor given as the colimit
over iterated suspension and loop space construction.
Then $(Q \Sigma^\infty X)_n \coloneqq Q(\Sigma^n X)$ is the $\Omega$-spectrum corresponding to the suspension spectrum of $X$.
The standard incarnation of the spectrum representing complex and real topological K-theory $K U$ and $K O$ is already an $\Omega$-spectrum, due to Bott periodicity
and
Frank Adams, part III, section 2 of Stable homotopy and generalised homology, 1974
Anthony Elmendorf, Igor Kriz, Peter May, section 1 of Modern foundations for stable homotopy theory, in Ioan Mackenzie James, Handbook of Algebraic Topology, Amsterdam: North-Holland (1995) pp. 213–253, (pdf)
Stanley Kochmann, section 3.3 of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996
Cary Malkiewich, section 2.2 of The stable homotopy category, 2014 (pdf)