Compound symmetry is essentially the "exchangeable" correlation structure, except with a specific decomposition for the total variance. For example, if you have mixed model for the subject $i$ in cluster $j$ response, $Y_{ij}$, with only a random intercept by cluster
$$ Y_{ij} = \alpha + \gamma_{j} + \varepsilon_{ij} $$
where $\gamma_{j}$ is the cluster $j$ random effect with variance $\sigma^{2}_{\gamma}$ and $\varepsilon_{ij}$ is the subject $i$ in cluster $j$ "measurement error" with variance $\sigma^{2}_{\varepsilon}$ and $\gamma_{j}, \varepsilon_{ij}$ are independent. This model implicitly specifies the compound symmetry covariance matrix between observations in the same cluster:
$$ {\rm cov}(Y_{ij}, Y_{kj}) = \sigma^{2}_{\gamma} + \sigma^{2}_{\varepsilon} \cdot \mathcal{I}(k = i) $$
Note that the compound symmetry assumption implies that the correlation between distinct members of a cluster is $\sigma^{2}_{\gamma}/(\sigma^{2}_{\gamma} + \sigma^{2}_{\varepsilon})$.
In "plain english" you might say this covariance structure implies that all distinct members of a cluster are equally correlated with each other and the total variation, $\sigma^{2} = \sigma^{2}_{\gamma} + \sigma^{2}_{\varepsilon}$, can be partitioned into the "shared" (within a cluster) component, $\sigma^{2}_{\gamma}$ and the "unshared" component, $\sigma^{2}_{\varepsilon}$.
Edit: To aid understanding in the "plain english" sense, consider an example where individuals are clustered within families so that $Y_{ij}$ denotes the subject $i$ in family $j$ response. In this case the compound symmetry assumption means that the total variation in $Y_{ij}$ can be partitioned into the variation within a family, $\sigma^{2}_{\varepsilon}$, and the variation between families, $\sigma^{2}_{\gamma}$.
