I knew how to encrypt or sign messages, but only from non-custodian wallets like electrum. How does this differ from that encryption? Why is PGP encryption needed?
PGP was originally designed so that people could send each other encrypted email without having to agree on passwords to use. Each person keeps a public key and a private key for themselves, and a short fingerprint can be derived from the public key to save space (like bitcoin addresses).
You can encrypt a message with your private key, and specify which fingerprints are allowed to decrypt your message. This data is encoded into the encrypted message which prevents people without the private key of the fingerprints from reading the contents (that is why OgNasty could not read my first message).
Think about it - if you just sent someone an encrypted text with just AES256 or something, how will you securely communicate the password that decrypts the text to the other party without compromising it? PGP solves this by not requiring you to send any secrets.
Also, I don't understand why we need a server that keeps the fingerprints (with our info, like name and email). Couldn't we just use the fingerprint of a person instead of searching the server for his name?
Key servers are convenient for importing someone's fingerprint from the internet instead of having to search for their public key (you need the public key to import a fingerprint). You could also just download the public key file yourself and import that.
Can someone proves me that he owns a fingerprint with digital signature? Because I see that you can only prove you own a public key that way.
Because a fingerprint is just a shorthand for of the public key, proving ownership of the public key also proves ownership of the fingerprint. If someone is able to sign a message from a particular PGP key, it only proves they have access to its private key. It does not prove that other people do not have access to it (sole ownership). You'd have to rely on their word for that.