Single localized polaron (quasiparticle) States are considered in structures relating to carbon nanotubes. The hamiltonian is derived in the tight-binding approximation first on a hexagonal lattice and later on a general carbon nanotube with specifiable chirality, and shares close links with the Davydov model of excitations of a one-dimensional molecular chain. First-order interactions of the lattice degrees of freedom with the electron on-site and exchange terms are included. The system equations are shown, under certain approximations, to share a close relationship with the nonlinear Schrödinger equation - an equation that is known to possess localised solutions. The ground state of system is investigated numerically and is found to depend crucially upon the strengths of the electron-phonon interactions.