Learn Russian: Сын очень похож на отца. - The son looks very much like the father.

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

If на takes the Accusative case here, why does отца look like it is in the Genitive case?

You are right that it looks like the Genitive! Because отец (father) is an animate masculine noun, its Accusative form is identical to its Genitive form. Grammatically, it is still the Accusative case here.

Why do we use the short-form adjective похож instead of a regular long-form adjective?

The short form похож is used to express a state or make a direct comparison as the main action of the sentence (the son is similar). You would use the long form похожий to describe a noun directly, like in the phrase похожий человек (a similar person).

What happened to the 'е' in the word отец when it became отца?

The word отец has a fleeting vowel. This is a common Russian spelling rule you might remember from earlier where the vowel е or о in the last syllable drops out when a case ending is added. That is why we get отца instead of 'отеца'.