Learn Russian: Сын похож на отца. - The son looks like his father.

The short-form adjective похож (looks like / is similar to) pairs with the preposition на followed by the Accusative case to indicate who or what someone resembles.

Why is the word for father отца here instead of отец?

Because похож на requires the Accusative case. Since отец (father) is a living person (an animate masculine noun), its Accusative form looks exactly like its Genitive form: отца.

Could I use the long-form adjective похожий instead of похож in this sentence?

Not here. Short-form adjectives like похож are used to make statements where the adjective acts as the main verb phrase (the predicate). You would use the long form похожий to describe a noun directly, like in the phrase похожий человек (a similar person).

The English translation says "his father," but there is no word for "his" in the Russian sentence. Why?

In Russian, possessives like свой or его are often left out when the context makes it obvious whose relative is being discussed. It is naturally assumed the son looks like his own father.