The electronic structures of a series of carbon nanotubes with different sizes, chiralities, ends, and bond lengths are studied systematically. Their absorption spectra are calculated with the localized density matrix method. The semiempirical model PM3 is employed in the calculation. The nature of optical excitations is investigated by examining their reduced single-electron density matrices. It is found that the optical excitations may be divided into the end modes and the tube modes, which have distinctive energies, features, and structural dependences. The optical gaps of carbon nanotubes scale inversely with tube length. Finite optical gaps have been confirmed and determined for infinitely long carbon nanotubes. The densities of states of carbon nanotubes are calculated at the self-consistent Hartree-Fock level. The calculated absorption spectra and densities of states compare well to the experimental results.