A search for the Standard Model Higgs boson produced in association with a pair of top quarks, $t\bar{t}H$, is presented. The analysis uses 20.3 fb$^{-1}$ of $pp$ collision data at $\sqrt{s}=8$ TeV, collected with the ATLAS detector at the Large Hadron Collider during 2012. The search is designed for the $H \to b\bar{b}$ decay mode and uses events containing one or two electrons or muons. The Higgs boson is assumed to have a mass of 125 GeV. In order to improve the sensitivity of the search, events are categorised according to their jet and $b$-tagged jet multiplicities. A multivariate technique is used to discriminate between signal and background events, which are dominated by $t\bar{t}$+jets production. No significant excess of events above the background expectation is found and an observed (expected) 95 % confidence-level limit of 4.1 (2.6) times the Standard Model cross section is obtained. By performing a fit allowing for an arbitrary Higgs boson signal strength, the ratio of the measured signal strength to the Standard Model expectation is found to be $\mu = 1.7 \pm 1.4$.

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