Learn Icelandic: Bærinn er fyrir sunnan þorpið. - The town is south of the village.

The word suður means the general direction south, like when you are traveling south. But to specify that something is south of a specific place, you need the preposition fyrir plus the special location adverb ending in -an, which gives us fyrir sunnan.

Why is þorpið in the accusative case?

Complex spatial prepositions made of fyrir plus an -an direction word always force the following noun into the accusative case. Because þorp is a neuter noun, its accusative definite form looks exactly the same as its nominative definite form.

Are bær and þorp synonyms, or is there a difference?

They are similar but not identical. A þorp is a smaller village, whereas a bær is usually a larger town. In other contexts, bær can also refer to a traditional Icelandic farm.

I have also seen sunnan við. Could we use that in this sentence?

Yes, you could say Bærinn er sunnan við þorpið. The phrases fyrir sunnan and sunnan við mean exactly the same thing and both require the following noun to be in the accusative case.