"That is my horse."
Translation:Dies ist mein Pferd.
I think I figured it out. Dieses is used if it directly modifies Pferd: Dieses Pferd ist schön. And given the context (where the horse is implied and understood by the listener), one could also say: Dieses ist schön. But in the sentence above, Dies serves as a separate, stand-alone word that points to the horse; it doesn't modify it. In such a case, Dies is not declined/adjusted.
As I understand it (and I've only been working on German for about 2 months), the form of the possessive pronoun mine depends on the noun it is modifies, and the form it takes parallels the form of the indirect article ein. Hence, ein Hund => mein Hund; einen Apfel => meinen Apfel. So... mein would be used when ein would be correct, and meine would be used when eine would be correct. In this particular sentence, Pferd is neuter noun, so it would be mein Pferd. If it were plural, then it would be meine Pferde.