Keys are made by your browser's Web Crypto API (crypto.subtle.generateKey) with public exponent 65537. Nothing is sent anywhere and nothing is saved. Generating a 4096-bit key can take a few seconds.
PKCS#8 / SPKI (BEGIN PRIVATE KEY / BEGIN PUBLIC KEY) is what most libraries expect. PKCS#1 (BEGIN RSA PRIVATE KEY) is the older RSA-only form. OpenSSH gives an ssh-rsa … line for authorized_keys and a private key that ssh accepts (save it as ~/.ssh/id_rsa and chmod 600 it). JWK is the JSON form used by JOSE / JWT libraries.
The private key is not encrypted. To add a passphrase: openssl pkey -in private.pem -aes256 -out private-enc.pem or ssh-keygen -p -f id_rsa.
Shortcut: Ctrl/⌘+Enter copies the public key.