SSH (Secure Shell) is how you get a secure remote shell, copy files (scp/sftp), or tunnel other traffic over an encrypted channel. Under the hood, it does three distinct jobs: encrypt the session, verify the server, and authenticate the client.
"SSL" is the name everyone still uses; the actual protocol in use today is TLS (SSL was its predecessor and is considered broken/deprecated). The handshake is what happens before any encrypted application data flows — it establishes who you're talking to and what key you'll both use to protect the conversation.