How to use:
The other party's public key:
If you want to encrypt something that only the other party (and no one else) can decrypt, you have to encrypt it with their public key (not with your key).
If you get a signature from the other party, you can verify that the signature is correct (as opposed to created by someone else) by using the other party's public key.
Your own private key:
If you want to sign something so that everyone can verify that you created the contents, you sign it with your own private key. Your public key will be used to verify it. The contents are not encrypted at all (unless you do that separately).
If someone sends you a message encrypted with your public key (so that only you can read it), you can decrypt it with your private key.
Your own public key:
You do not use your own public key. The other party uses it to verify your signatures, and to encrypt messages for your eyes only.
The other party's private key:
You do not have that.