Learn Hardcore Latin: Auriga calcar habet.

auriga is in the nominative singular, the case normally used for the subject.
calcar is the direct object, so it is in the accusative singular.

One small complication: calcar is a neuter noun, and in Latin neuter nouns often have the same form in the nominative and accusative singular. So here the form calcar could look like either case by itself. In this sentence, the normal meaning and word order make it clear that auriga is the one who has something, and calcar is the thing had.

Why does auriga end in -a if it refers to a man?

Because auriga is a masculine noun of the first declension.

English speakers often expect nouns ending in -a to be feminine in Latin, and many are. But some first-declension nouns referring to male professions or roles are masculine, such as:

agricola — farmer
nauta — sailor
poeta — poet
auriga — charioteer

So the ending tells you the declension, not always the natural gender.

What form is habet?

Habet is the of , meaning .