In a key pair, what the public key encrypts, only the private key can decrypt, and vice versa. I’ve never heard or a backup public key in such a scenario, though I suppose its technically possible. Foolish, though, but that’s outside the scope of your question.
Assuming your “backup” public key is actually THE public key, you would give the public key to people you want to be able to decrypt something you encrypt with the private key, like an email (key pairs aren’t suitable for encrypting large files, they’re too slow) or to verify that you digitally signed a document. The reverse is also possible if they use your public key to encrypt or sign something, then only you can decrypt or verify it.
Finally, it is important to remember that whatever your private key encrypts, all copies of the public key can decrypt (so whomever you’ve given it to, be it 1 person or 1000). If you want a private email conversation, for example, you exchange public keys with the other party. Then you encrypt an email you send with their public key (because there should only ever be one copy of the private key, held by the owner) so that only they can decrypt it, then they’ll do the same thing to their response with your public key so only you can decrypt it. In this way you have secure communication.
I hope that’s clear enough, please let me know if it isn’t.