"She likes his dog."
Translation:Sie mag seinen Hund.
It's similar to the difference between der/die/das/den/etc. - they are different forms of the same word, modified according to gender (masculine, feminine, neuter or plural) and case (nominative, accusative, dative and genitive). This is something that happens in German but not in English. This 'modification' is called declension.
All the possessive determiners ("his", "her", "its", "their", "our", "your", "my") in German follow the same declension pattern for the indefinite article ("a"/"an"), i.e. for ein/eine. You'll need to memorise that.
Here you've got a masculine noun in accusative case, which corresponds to the -en ending for the indefinite article: ...seinen Hund. If you wanted to say "...my dog" it'd be ...meinen Hund and for "...his car" it'd be ...sein Auto and so on.
Does that explain to you why there are different endings? The meaning of them is all basically the same. Seins is a bit different, as it's a pronoun (it replaces a neuter noun, basically meaning "his one").