Literally, 'the description or study of appearances'.
Any detailed study of a phenomenon can be called a phenomenology, but the theory normally so called is associated with Franz Brentano (1838-1917) and (especially) Edmund Husserl (1859-1938) and their followers, including several existentialists.
'Phenomena' for Husserl were the objects of experience or attitudes (in the sense in which even a non-existent fortune can be the object of my wish). These he treated as essences, aiming to give an analysis of them not unlike the 'conceptual analysis' of linguistic philosophy.
The analysis, however, was a priori, and he aimed (at any rate in his later works) to avoid psychologism, laying aside (bracketing) ideas derived from empirical science.
Also see: linguistic phenomenology
H Spiegelberg, The Phenomenological Movement (1960)