We present a theoretical investigation of electron states hosted by magnetic domain walls on the three-dimensional topological insulator surface. The consideration includes the domain walls with distinct vectorial and spatial textures. The study is carried out on the basis of the Hamiltonian for quasirelativistic fermions by using a continual approach and tight-binding calculations. We derive the spectral characteristics and spatial localization of the one-dimensional low-energy states appearing at the domain walls. The antiphase domain walls are shown to generate the topologically protected chiral states with linear dispersion, the group velocity and spin-polarization direction of which depend on an easy-axis orientation. In the case of an easy-plane anisotropy, we predict a realization of a dispersionless state flatband in the energy spectrum, that is spin polarized along the surface normal. Modification of the surface states in the multidomain case, which is approximated by a periodic set of domain walls, is described as well. We find that the magnetic domain walls with complex internal texture, such as Ned-like or Bloch-like walls, also host the topological states, although their spectrum and spin structure can be changed compared with the sharp wall case.