Generate a key pair with ssh-keygen. Modern standard is Ed25519 (fast, secure, small keys). Older RSA keys are also supported but larger.

$ ssh-keygen -t ed25519 -C "user@host"
$ ssh-keygen -t rsa -b 4096 -C "user@host"     # older standard

This creates ~/.ssh/id_ed25519 (private key) and ~/.ssh/id_ed25519.pub (public key). Protect the private key with a passphrase. The passphrase encrypts the private key file—if someone steals it, they can't use it without the passphrase.

Generating public/private ed25519 key pair.
Enter file in which to save the key (/home/user/.ssh/id_ed25519): 
Enter passphrase (empty for no passphrase): 
Enter same passphrase again: 

The public key is safe to share. Copy it to servers and add to ~/.ssh/authorized_keys:

$ ssh-copy-id -i ~/.ssh/id_ed25519.pub user@host

Or manually:

$ cat ~/.ssh/id_ed25519.pub | ssh user@host "cat >> ~/.ssh/authorized_keys"

File permissions matter:

$ chmod 700 ~/.ssh              # directory
$ chmod 600 ~/.ssh/id_ed25519   # private key
$ chmod 644 ~/.ssh/id_ed25519.pub  # public key
$ chmod 644 ~/.ssh/authorized_keys  # on server

View key fingerprint: ssh-keygen -lf ~/.ssh/id_ed25519.pub. Never share your private key.