We review quantum-chemical studies of the excited-state electronic structure of finitesize semiconducting single-walled carbon nanotubes (SWCNTs) using methodologies previously successfully applied to describe conjugated polymers and other organic molecular materials. The results of our simulations are in quantitative agreement with available spectroscopic data and show intricate details of excited-state properties and photoinduced vibrational dynamics in carbon nanotubes. We analyze in detail the nature of strongly bound first and second excitons in SWCNTs for a number of different tubes, emphasizing emerging size-scaling laws. Characteristic delocalization properties of excited states are identified by the underlying photoinduced changes in charge densities and bond orders. Due to the rigid structure, exciton-phonon coupling is much weaker in SWCNTs compared to typical molecular materials. Yet we find that, in the ground state, a SWCNT's surface experiences the corrugation associated with electron-phonon interactions. Vibrational relaxation following photoexcitation reduces this corrugation, leading to a local distortion of the tube surface, which is similar to the formation of self-trapped excitons in conjugated polymers. The calculated associated Stokes shift increases with enlargement of the tube diameters. Such exciton vibrational phenomena are possible to detect experimentally, allowing for better understanding of photoinduced electronic dynamics in nanotube materials.