The scam is often advanced fees. Usually they want you to pay transportation fees. You can pay those but you will never get the puppy. There is no puppy.
If they are sending you an animal to take care of they may want to pay you for several months boarding and they will overpay with a check. They will want you to send the balance back to them by wire transfer (Western Union of course) so that is a fake check scam because the check is always fake.
Silence is the scammer's best friend; knowledge is the scammer's worst enemy. 沈黙は詐欺師のよき友達、知識は詐欺師の天敵。Think globally, act globally.