Well, the meaning: "mój" = "my", "twój" = "your (thy)" (the possesive form of "you, y'all" is "wasz" (for masculine nouns in the nominative)).
"Mój koń" = "My horse", "Twój koń" = "Your horse" (horse is a masculine noun in Polish, the case used here is the nominative)
"Moja krowa" = "My cow", "Twoja krowa" = "Your cow" (cow is a feminine noun in Polish)
"Widzę mojego konia" = "I see my horse" (here you use a different case and the words are inflected ("widzę mój koń" is incorrect!))
The grammatical number. In English, when you say 'you' or 'your' it does not show whether you mean singular 'you' or plural 'you'. Polish shows it.
Therefore: you will use "twój" if the horse belongs to Adam and you're talking to him. You will use "wasz" if the horse belongs to Adam and Julia (even if you're just talking to Adam, you still may easily use 'wasz' because you mean both him and Julia).