Plasmonic modes at mid-infrared wavelengths in elliptical graphene disk arrays were studied. Theoretically, analytical expressions for the modes and their dependence on the size, Fermi energy and the permittivity of substrate materials of the ellipses were derived. Experimentally, the elliptical graphene disks were fabricated and their plasmonic modes were characterized with the polarization-resolved extinction spectra. Both experimental and analytical results show that two electrical dipole modes, whose dipole moments are orthogonal to each other and along the major and minor axis of the ellipse respectively, exist in the elliptical disks. By adjusting the polarization directions of the incident light, the two orthogonal plasmonic modes could be excited either together or separately, showing that the optical properties of elliptical graphene disks are highly polarization dependent. By using ultraviolet illumination to change the Fermi energy of the elliptical graphene disks, the two modes can be tuned dynamically. Moreover, the highly polarization dependent modes are able to couple with the surface phonons of the substrate, leading to polarized plasmon-phonon polaritons. Thus the elliptical graphene disks can provide more degrees of freedom to design the mid-infrared polarization-resolved photonic devices.