When you use Signal, your data is stored encrypted on your devices. All of it. Signal servers only get the phone number you joined with, when you joined and last logged on.
The Signal app uses the Signal protocol. If you want the technical side of things, the protocol is built on the Double Ratchet algorithm, triple Elliptic-curve Diffie–Hellman (3-DH) handshake, and prekeys. It uses Curve25519, AES-256, and HMAC-SHA256 for its encryption algorithms.
Interestingly, apps like WhatsApp use the Signal protocol partially, to encrypt message content. WhatsApp chooses to collect your metadata for its own use (they pass it along to Facebook too). More here.
Don’t even get me started on email. PGP must be over 20 years old now.