A formula for the relaxation time of electrons at the surface of a sphere consisting of a metal core and a metal shell (core-shell) has been derived and the frequency dependences of the electric and magnetic components of the sphere polarizability have been analyzed. The sphere polarization maxima correspond to the plasmon resonances of the core and the shell. Calculations are carried out for bimetallic Ag–Au, Au–Ag, and Ag–Al nanoparticles embedded in Teflon. A possibility to control the optical characteristics of bimetallic nanoparticles by changing their morphology is demonstrated. The extinction and scattering cross-sections and the field enhancement factor in the nanoparticle vicinity are calculated in a wide spectral interval and for various core-to-shell size ratios. The temperature of a photothermally heated bimetallic nanosphere used for the therapy of malignant tumors is evaluated.