Originally it used to be Alma Ata which then became Almatı. This kind of transformation is common in Turkic languages.
Ata does mean father or forefather or grandfather. It is also used in the sense of "ancestor" in a similar sense.
In that sense, it is like көке (köke) which is derived from the word "root". It is not used in Turkey like that, but any Turk would understand why it means "father".
Depends on what you mean by "originally" ;)
The modern city was founded in 1854 when the Russians established the military fortification of Zailiyskoye (renamed Verny in 1855) on the site of the ancient settlement of Almaty, which had been destroyed by the Mongols in the 13th century.
In Soviet time (1921) the city was indeed renamed to "Alma-Ata", but only in Russian, in Kazakh the name became "Almaty". In 1993, the government renamed the city from the Russian "Alma-Ata" to the Kazakh name "Almaty".
In any case, it is kind of incorrect to translate "Alma-Ata" from Kazakh, because this name was never used in that language.