  We report on measurements of dielectron ($e^+e^-$) production in Au$+$Au
collisions at a center-of-mass energy of 200 GeV per nucleon-nucleon pair using
the STAR detector at RHIC. Systematic measurements of the dielectron yield as a
function of transverse momentum ($p_{\rm T}$) and collision centrality show an
enhancement compared to a cocktail simulation of hadronic sources in the low
invariant-mass region ($M_{ee}<$ 1 GeV/$c^2$). This enhancement cannot be
reproduced by the $\rho$-meson vacuum spectral function. In minimum-bias
collisions, in the invariant-mass range of 0.30 $-$ 0.76 GeV/$c^2$, integrated
over the full $p_{\rm T}$ acceptance, the enhancement factor is 1.76 $\pm$ 0.06
(stat.) $\pm$ 0.26 (sys.) $\pm$ 0.29 (cocktail). The enhancement factor
exhibits weak centrality and $p_{\rm T}$ dependence in STAR's accessible
kinematic regions, while the excess yield in this invariant-mass region as a
function of the number of participating nucleons follows a power-law shape with
a power of 1.44 $\pm$ 0.10. Models that assume an in-medium broadening of the
$\rho$ meson spectral function consistently describe the observed excess in
these measurements. Additionally, we report on measurements of $\omega$ and
$\phi$-meson production through their $e^+e^-$ decay channel. These
measurements show good agreement with Tsallis Blast-Wave model predictions as
well as, in the case of the $\phi$-meson, results through its $K^+K^-$ decay
channel. In the intermediate invariant-mass region (1.1$<M_{ee}<$ 3 GeV/$c^2$),
we investigate the spectral shapes from different collision centralities.
Physics implications for possible in-medium modification of charmed hadron
production and other physics sources are discussed.
