The Lyman-alpha transition occurs when an electron in a hydrogen atom transitions from the first excited state (n=2) to the stable ground state (n=1), emitting an ultraviolet photon at 1215.67 Å. This and the other Lyman transitions to the ground state are named after American physicist and spectroscopist Theodore Lyman (1874-1954) who discovered and studied these spectral lines.
About 75% of the mass of our universe is hydrogen, so when we look at a very distant object, such as a quasar, the light from that distant object passes through a large number of tenuous hydrogen clouds between us and the distant object. The cooler hydrogen clouds absorb ultraviolet light at a wavelength of 1215.67 Å, so this wavelength is “removed” from the light from a distant object, as evinced by an absorption line in the spectrum of the distant object. But because the intervening neutral hydrogen clouds are moving at different speeds and cosmological redshifts, a number of different wavelengths have light removed (as seen from Earth), resulting in what is known as a Lyman-alpha forest. Analysis of the Lyman-alpha forest can tell us much about the neutral hydrogen clouds between us and any distant extragalactic source.
When a hydrogen cloud atom absorbs a 1215.67 Å ultraviolet photon, its electron jumps from the n=1 ground state up to the n=2 first excited state. However, excited electrons can’t stay in the n=2 state for long, and quickly return to the ground state again, emitting a photon of light at 1215.67 Å. So, why do we even see an absorption line? Yes, ultraviolet photons from the distant extragalactic source are removed from our line of sight by an intervening hydrogen cloud, but when ultraviolet photons are re-emitted, the photons radiate in all directions, and only a few travel towards us along our line of sight. The net result is an absorption line.