Learn Icelandic: Ég heyri hljóð lamba og svína. - I hear the sounds of lambs and pigs.

They are in the genitive plural, which translates to "of lambs" and "of pigs". For bare nouns (nouns without "the" attached to the end), the genitive plural almost always ends in -a. So, lamb becomes lamba and svín becomes svína.

The English translation says "sounds", so why doesn't hljóð have a plural ending?

Hljóð is a neuter noun. In Icelandic, neuter nouns that are just one syllable long look exactly the same in the plural as they do in the singular for the nominative and accusative cases. So, hljóð can mean "a sound" or "sounds".

Does the verb heyra (to hear) affect the case of hljóð?

Yes, heyra causes the thing you hear to be in the accusative case. However, because hljóð is a neuter word, its accusative form is identical to its dictionary (nominative) form, so you don't see a visible change here.

Could I put the animals first, like "lamba og svína hljóð"?

While English usually puts the possessor first ("lambs' sounds"), Icelandic strongly prefers to put the possessed object first, followed by the owners in the genitive case. It literally flows as "I hear sounds of lambs and pigs".