Azureus uses public key cryptography to ensure that communication between friends/buddies is confidential.
Each Azureus install has a separate key pair - a public key and a private key.
The private key is only ever stored on the local disk of your machine.
The public key can be communicated to the world at large to allow people to securely communicate with you.
It is important that the private key is kept securely. To this end it is stored in an encrypted form itself using a password. The management of this password can be by the user themselves or automatically by the client (the default). In both cases the private key never leaves your machine, it is purely the management of the password required to unlock it for use that differs.
I keep getting a pop up box asking for an encryption key password?
Somehow you've managed to set a password for your private key... Short of remembering the actual password, you can regenerate a new one:
Tools - Options - Mode: select Advanced
Then go to the Security section, and you should see a Reset (keys) button. Click it and confirm the warning. You might be prompted for a new password. Once you've recreated the Public/Private keys, exit out of the client and restart it.