Single-wall carbon nanotubes (SWCNTs) are unique one-dimensional (1D) condensed matter systems in which strongly enhanced Coulomb interactions are combined with unusual band structure. There are metallic and semiconducting SWCNTs, in both of which electron-electron interactions have significant impact on their electronic and optical properties. In this dissertation work, we used ultrafast optical pump-probe spectroscopy to investigate nonequilibrium dynamics of photogenerated electron-hole pairs, or excitons, in a sample in which a particular species, or chirality, of semiconducting SWCNTs was enriched. Specifically, we studied both an aqueous suspension and an aligned film of (6,5) SWCNTs. Depending on the pump photon energy, intensity, and polarization, different physical processes ensue after ultrafast pumping, including coherent light-matter interactions and incoherent relaxation of carriers/excitons. For example, under below-gap pumping, a transient blueshift of the exciton peak occurred, only during the pump pulse duration, a hallmark of the optical Stark effect. Under resonant pumping, transient splitting of the exciton peak was observed within the pulse duration, which is a manifestation of the Rabi doublet due to coherent light-matter interaction in the strong coupling regime. The Rabi doublet was observed only under resonant or near-resonant pumping conditions. In the case of a macroscopically aligned (6,5) SWCNT film sample, an anisotropic Rabi doublet of the exciton peak was observed under resonant pumping. In the case of above-gap excitation, incoherent relaxation processes dominated the dynamics of excitons. Analysis of these ultrafast, nonequilibrium, and strongly driven phenomena provided considerable new insight into the states and dynamics of electrons in the presence of extreme quantum confinement and strong many-body interactions.