A measurement of spin correlation in $t\overline{t}$ production is presented using data collected with the ATLAS detector at the Large Hadron Collider in proton-proton collisions at a center-of-mass energy of 8 TeV, corresponding to an integrated luminosity of $20.3\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$. The correlation between the top and antitop quark spins is extracted from dilepton $t\overline{t}$ events by using the difference in the azimuthal angle between the two charged leptons in the laboratory frame. In the helicity basis the measured degree of correlation corresponds to ${A}_{\text{helicity}}=0.38\ifmmode\pm\else\textpm\fi{}0.04$, in agreement with the standard model prediction. A search is performed for pair production of top squarks with masses close to the top quark mass decaying to predominantly right-handed top quarks and a light neutralino, the lightest supersymmetric particle. Top squarks with masses between the top quark mass and 191 GeV are excluded at the 95% confidence level.