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 STARs 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.
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