Very roughly speaking, a spectrum is an object that has stable homotopy groups, and the sphere spectrum is the spectrum whose stable homotopy groups are those of the spheres.
The starting point is the Freudenthal suspension theorem.
First, recall that for any pointed space $X$, we have a natural map from $X$ to $\Omega \Sigma X$, the loop space of the suspension of $X$.
(Explicitly, if we think of $\Sigma X$ as a quotient of $[0, 1] \times X$, the natural map $X \to \Omega \Sigma X$ sends each point $x$ to the loop defined by $t \mapsto (t, x)$.)
Thus, for every $k$, we have a natural homomorphism $\pi_k (X) \to \pi_k (\Omega \Sigma X)$; but we also have a natural isomorphism $\pi_k (\Omega \Sigma X) \cong \pi_{k+1} (\Sigma X)$, so by composition we get a natural homomorphism $\pi_k (X) \to \pi_{k+1} (\Sigma X)$.
The theorem implies that, for any pointed CW complex $X$, for all $k$ and all sufficiently large $n$, the natural homomorphism $\pi_k (\Sigma^n X) \to \pi_{k+1} (\Sigma^{n+1} X)$ is an isomorphism.
In other words, the sequence
$$\pi_k (X) \longrightarrow \pi_{k+1} (\Sigma X) \longrightarrow \pi_{k+2} (\Sigma^2 X) \longrightarrow \cdots$$
eventually stabilises.
The $k$-th stable homotopy group of the spheres is the "eventual value" of the sequence for $X = S^0$.
(Recall that $\Sigma^n S^0 \cong S^n$.)
More generally, instead of considering the sequence of pointed spaces $(X, \Sigma X, \Sigma^2 X, \ldots)$ and maps $\Sigma^n X \to \Omega \Sigma^{n+1} X$, we may consider a general sequence of pointed spaces $(X_0, X_1, X_2, \ldots)$ with maps $X_n \to \Omega X_{n+1}$.
This is the data of a spectrum (perhaps in a broad sense – definitions vary), and from the earlier discussion we see that we get a sequence
$$\pi_k (X_0) \longrightarrow \pi_{k+1} (X_1) \longrightarrow \pi_{k+2} (X_2) \longrightarrow \cdots$$
though it may or may not eventually stabilise; regardless, we may define the $k$-th stable homotopy group to be $\varinjlim_n \pi_{k+n} (X_n)$.
(For $k \le 0$, we skip the first $1 - k$ steps of the sequence, of course.)
If the given maps $X_n \to \Omega X_{n+1}$ are (weak) homotopy equivalences for all sufficiently large $n$, then the sequence does eventually stabilise.
An $\Omega$-spectrum is a spectrum where these maps are (weak) homotopy equivalences for all $n$.
Not every spectrum is an $\Omega$-spectrum, but there is a universal way of replacing any spectrum with an $\Omega$-spectrum that has the same stable homotopy groups.
Essentially, we replace $X_k$ with $\varinjlim_n \Omega^n X_{k+n}$.
The evident maps $X_k \to \varinjlim_n \Omega^n X_{k+n}$ assemble to yield a morphism of spectra, and the induced homomorphism of stable homotopy groups is an isomorphism – i.e. we have a stable weak homotopy equivalence.
The stable homotopy category is the result of localising the category of spectra with respect to stable weak homotopy equivalence.
Notice that a morphism of $\Omega$-spectra is a stable weak homotopy equivalence if and only if all of its components are weak homotopy equivalences – this fact makes $\Omega$-spectra more technically convenient than general spectra, at least in some contexts.
As alluded to in the comments, there is not yet a universally accepted definition of "spectrum".
However, there is agreement on what the stable homotopy category "is", and all reasonable definitions of "spectrum" yield the same stable homotopy category (up to equivalence, of course).
The stable homotopy category has a symmetric monoidal closed structure where the monoidal product is induced by the smash product, and good definitions of "spectrum" yield a category of spectra with a symmetric monoidal closed structure that descends to that of the stable homotopy category.
Not all definitions of "spectrum" are good – for some definitions, it is mathematically impossible to obtain such a symmetric monoidal closed structure.
Thus, if we do not commit to a particular good definition of "spectrum" there may not even be a symmetric monoidal category of spectra to speak of!
One may try to avoid this problem by working with the stable homotopy category instead, and indeed the sphere spectrum $(S^0, S^1, S^2, \ldots)$ is the unit of the monoidal structure on the stable homotopy category, but I think this is a poor definition because it does not distinguish between stably weakly homotopy equivalent spectra.
However, I am not a practitioner, so maybe I have the wrong perspective here.
