The word "background" is essentially jargon used to describe the zeroth order terms in perturbation theory.
In general, you can write the metric and stress-energy tensor as
\begin{eqnarray}
g_{\mu\nu} &=& \bar{g}_{\mu\nu} + \epsilon h_{\mu\nu} \\
T_{\mu\nu} &=& \bar{T}_{\mu\nu} + \epsilon \tau_{\mu\nu}
\end{eqnarray}
Then the "background" equations of motion are the Einstein equations and conservation of stress-energy when $\epsilon=0$
\begin{eqnarray}
\bar{G}_{\mu\nu} &=& 8 \pi G_N \bar{T}_{\mu\nu} \\
\bar{\nabla}_\mu \bar{T}^{\mu\nu}&=&0
\end{eqnarray}
So far it probably looks like we haven't done much, but the point of perturbation theory is to start with a background $\bar{g}_{\mu\nu}$ that you know solves the Einstein equations. In cosmology, this indeed means that the background metric is an FLRW metric (with an appropriate isotropic and homogenous stress-energy describing the matter).
The first order perturbation equations then can be computed by plugging the form of the metric and stress-energy tensor into Einstein's equations and expanding to linear order in $\epsilon$. However, I should warn you, that these equations are a huge mess in general.
On the other hand, because there is a lot of symmetry in the FLRW background (and, realistically, a lot of symmetry in any exact background solution you're likely to come across), you can do better than just plugging $\bar{g}_{\mu\nu} + \epsilon h_{\mu\nu}$ directly into Einstein's equations. Typically you decompose the metric perturbation $h_{\mu\nu}$ and matter perturbation $\tau_{\mu\nu}$ into invariant components based on the symmetries of the background.
For FLRW, we make use of the rotational symmetry of the background. This lets us can decompose the metric perturbation into 4 scalar components, 2 transverse (spatial) vector components, and a transverse-traceless (spatial) tensor component. As a sanity check, note 4 scalars = 4 degrees of freedom, 2 transverse vectors = 4 degrees of freedom, and a transverse-traceless tensor = 2 degrees of freedom, and 4+4+2=10 is the right number of degrees of freedom for the metric perturbation $h_{\mu\nu}$. There are additionally 4 gauge symmetries associated with general coordinate invariance that you will fix to reduce the number of degrees of freedom; exactly how this is done depends on what you want to do, and different books will make different choices, so I am just flagging that this will come up and will leave it to your book to explain it in more detail. One major advantage of this decomposition is that (to order $\epsilon$) the equations for the scalar, vector, and tensor components decouple, which means they can be treated independently.
