"That small animal is a dog."
Translation:Tamto małe zwierzę to pies.
That's just a way of creating sentences of the "X is Y" type, with both X and Y usually being nouns or noun phrases. They're both used in Nominative.
- Pies to zwierzę (A dog is an animal).
- Twój tata to miły człowiek (Your father is a nice man)
- Poprawna odpowiedź to "Czechy" (The correct answer is "Czech Republic")
But not only:
- Zielony to mój ulubiony kolor (Green is my favourite colour) - this could go the other way round as well.
This does not go well with using pronouns ("On to lekarz" would be a rather unnatural sentence), so be careful.
Those constructions are equivalent and generally mean the same, although of course there are some shades of meaning, which are rather hard to explain. "to" is a bit like a "=" sign. "jest" is more... like a description. That animal belongs to the canine family.
In this particular sentence, I would choose "to". A simple "Yup, that's a dog". But on the learner's level, they're generally similar enough.