Christian Doppler described the effect in his treatise
“Über das farbige Licht der Doppelsterne und einiger
anderer Gestirne des Himmels—Versuch einer das
Bradley’sche Aberrations-Theorem als integrirenden Theil in
sich schliessenden allgemeineren Theorie” (On the coloured
light of the binary stars and some other stars of the
heavens—Attempt at a general theory including
Bradley’s theorem as an integral part), which has a fairly
descriptive title. He used the Doppler effect to explain the color
of binary stars.
Doppler is not recognized as the father of relativity.
Albert Einstein used the Doppler-shift factor to help establish
the equivalence of mass and energy in 1905 and reformulated the
Doppler effect to account for the time dilation effect of the
special theory of relativity.
See also in The book of science:
Readings on wikipedia: