A quenched galaxy is a galaxy with minimal star formation, i.e., the opposite of a starburst galaxy. They are determined by color: a lack of the blue evident of early (hot, short-lived) stars is consistent with a lack of recent star formation. An obvious cause is that the galaxy has little gas, specifically lacking molecular clouds, from which stars are assumed to be formed. This could be caused by a gas blowout, e.g., from an active galactic nucleus.
Another possible cause is that the molecular gas is too hot or turbulent (molecular cloud turbulence) to contract into stars, possibly the eventual outcome of a galaxy merger. A measure of the latter is whether the gas motion evident from Doppler shifts matches the stellar motion (misaligned gas).
The quenching process itself sometimes takes a long time (slow quenching, with a timescale of a gigayear or more) or a relatively short time (rapid quenching, 100My or smaller timescale). The slower process may be attributed to a reduction in gas entering the galaxy through some means, e.g., if it is blocked, or if something prevents the gas from cooling sufficiently.