euren is a form of the possessive pronoun that corresponds to the plural personal pronoun ihr (which is sometimes translated into English as you all or yous).
Basically, the dog belongs to at least two persons whom are directly addressed in the sentence above.
(Lukas likes the dog of you two / you all / yous.)
Just adding: euren is also the accusative form. In German, possessive pronouns have to agree in case with the noun that they accompany, just like the articles do.
i have a question the way ,'eu' is pronounced in the audio. Should it not be like ,'oi' as in english,'oil'? Could someone help me?