ssh is a protocol that is used to log in to a computer remotely. Servers are usually administrated not by plugging a keyboard and monitor into the server, but from another machine via ssh. You can configure ssh to allow login with the same username+password you would use locally, but it is common practice to only allow authentication with an ssh key.

ssh keys allow for much higher entropy like you suggested. They are also asymmetric, and the private key can be password-protected or stored on a smartcard.