"This dog is eating meat."
Translation:Ten pies je mięso.
To means both "it" and "this" in the gender-neuter case.
To jest pies. = "It is a dog." = "This is a dog."
Ten means "this" in the gender-masculine case.
Ten pies je mięso. = "This dog eats meat."
Pies ("dog") is a noun with a masculine case, so the noun determiner ten ("this") also takes the masculine case.
From what I picked up from some of the comments there are some rules of thumb, you can sometimes tell the gender by their last letter:
masculine nouns end with a consonant
female nouns end with -a
* neuter nouns end with -e, -ę, -o, -um.
Of course there are always exceptions from the rule, like "mężczyzna": it ends with an -a, but it's masculine.
Because mięso (don't forget the letter "Ę") is gender-neuter so the accusative case is the same as the nominative. If you were to say that this dog does not eat meat, then you would use the genitive case mięsa. In negation, the verb's object that would normally take the accusative case, takes the genitive case instead.
Ten pies nie je mięsa.
Because the object of the verb takes the accusative case. The dog is eating WHAT? Mięso. The accusative case of a gender-neuter noun is the same as the nominative case. Your example would be the instrumental case of mięso. The instrumental case would be, a man is eating WITH WHAT? WITH A FORK (widelec). Mężczyzna je widelcem.