A search for the standard model Higgs boson ($H$) decaying to $b\overline{b}$ when produced in association with a weak vector boson ($V$) is reported for the following channels: $W(\ensuremath{\mu}\ensuremath{\nu})H$, $W(e\ensuremath{\nu})H$, $W(\ensuremath{\tau}\ensuremath{\nu})H$, $Z(\ensuremath{\mu}\ensuremath{\mu})H$, $Z(ee)H$, and $Z(\ensuremath{\nu}\ensuremath{\nu})H$. The search is performed in data samples corresponding to integrated luminosities of up to 5.1 inverse femtobarns at $\sqrt{s}=7\text{ }\text{ }\mathrm{TeV}$ and up to $18.9\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ at $\sqrt{s}=8\text{ }\text{ }\mathrm{TeV}$, recorded by the CMS experiment at the LHC. An excess of events is observed above the expected background with a local significance of 2.1 standard deviations for a Higgs boson mass of 125 GeV, consistent with the expectation from the production of the standard model Higgs boson. The signal strength corresponding to this excess, relative to that of the standard model Higgs boson, is $1.0\ifmmode\pm\else\textpm\fi{}0.5$.