Learn Russian: Жених приехал на машине отца.

When talking about a mode of transportation, Russian almost always uses на followed by the Prepositional case. Even though English says "in a car" or "by car", Russian expresses this as traveling "on" the vehicle, hence на машине.

Why is the prefix при- used in the verb приехал?

The prefix при- adds the meaning of "arrival" to verbs of motion. Since ехать means to travel by vehicle, adding при- creates приехал, meaning "arrived by vehicle." If the groom had just departed, you would use a different prefix, like уехал (left/drove away).

How does отца translate to "his father's", and why does it come after the car?

Отца is in the Genitive case, which indicates possession (literally meaning "of the father"). In Russian, the noun showing who owns something comes immediately after the object being owned. Also, Russian often drops possessive pronouns like "his" when context makes it obvious whose father is being discussed.

Why did the letter е disappear when отец (father) changed to отца?

The word отец contains what is known as a "fleeting vowel." In many masculine Russian nouns, especially those ending in -ец or -ок, the vowel in the final syllable drops out when you add case endings. This simply makes the word easier to pronounce in fluid speech.