A single hydrogen atom has only one electron. Thus, it is in an (electronic) doublet state and there are no singlet or triplet states. The two states of the doublet are different in energy due to spin-orbit coupling. Basically, the electron can spin in the same direction as it is orbiting the nucleus or in the opposite direction. This is known as the fine structure and it is what your book refers to. This is not accounted for in the Bohr model because the Bohr model does not treat spin.

However, this is only the case if one neglects the spin of the nucleus. The spin of the nucleus can be parallel or anti-parallel to the spin of the electron leading to "triplet-like" or "singlet-like" states. The Bohr model does not account for this effect either. This is known as the Hyperfine structure.