What you'll do with the result
A CSR (Certificate Signing Request) is what you submit to a Certificate Authority to get an SSL/TLS certificate issued - it contains your public key and identifying details, but not your private key, which stays with you. Submit the CSR text to your CA, and keep the private key for installing alongside the certificate they send back. Since the key is generated fresh in this one request and never stored, generating a new CSR means generating a brand new key pair each time - you can't regenerate a lost key later.
Why the private key warning matters so much
The private key shown after generation exists only in that one response - it's never written to any database or file on this server, which is deliberate: a web tool that stored generated private keys would be a serious security liability. This also means there's no way to recover a lost key later, so copying and saving it immediately, before navigating away, is essential.
Choosing between 2048-bit and 4096-bit
2048-bit RSA is the current widely-accepted standard, fast to generate and supported everywhere. 4096-bit offers a theoretically stronger key at the cost of slower generation and slightly slower TLS handshakes, and isn't universally supported by older systems - most sites have no specific need to go beyond 2048-bit, which remains considered secure for the foreseeable future.
Filling in the organizational fields accurately
The Common Name must exactly match the domain the certificate will protect - a mismatch here is one of the most common reasons a CSR needs to be regenerated. Organization, locality, and other fields are typically required for higher-assurance certificate types (like Organization Validation) but can be left blank for a basic domain-validated certificate, depending on what your Certificate Authority specifically requires.