TL;DR
Depending on the specific SSH implementation, you can certainly specify multiple authentication mechanisms, and even control their order of precedence. With OpenSSH, the default is to try SSH keys before trying passwords, but the allowed authentication mechanisms and precedence can be modified in configuration files.
There are different settings related to both the client and the server. You may need to modify and reload/restart both for the any changes that implement your preferred set of options to properly take effect.
OpenSSH Configuration Options
Client Configuration
This is usually stored in ~/.ssh/config on *nix systems, and controls what authentication settings your SSH client will attempt. Some of the relevant settings include:
PreferredAuthentications
Specifies the order in which the client should try authentication methods. This allows a client to prefer one method (e.g. keyboard-interactive) over another method (e.g. password). The default is:
gssapi-with-mic, hostbased, publickey, keyboard-interactive, password
PasswordAuthentication
Specifies whether to use password authentication. The argument to this keyword must be yes (the default) or no.
Others may apply as well. See man 5 ssh_config for a complete list.
Server Configuration
This is generally stored in /etc/ssh/sshd_config, but the actual location may be distribution or OS specific. This file controls what authentication settings the server will allow. Some of the essential configuration items include the following.
AuthenticationMethods
Specifies the authentication methods that must be successfully completed for a user to be granted access. This option must be followed by one or more lists of comma-separated authentication method names, or by the single string any to indicate the default behaviour of accepting any single authentication method. If the default is overridden, then successful authentication requires completion of every method in at least one of these lists.
For example, "publickey,password publickey,keyboard-interactive" would require the user to complete public key authentication, followed by either password or keyboard interactive authentication. Only methods that are next in one or more lists are offered at each stage, so for this example it would not be possible to attempt password or keyboard-interactive authentication before public key.
For keyboard interactive authentication it is also possible to restrict authentication to a specific device by appending a colon followed by the device identifier bsdauth, pam, or skey, depending on the server configuration. For example, "keyboard-interactive:bsdauth" would restrict keyboard interactive authentication to the bsdauth device.
If the publickey method is listed more than once, sshd(8) verifies that keys that have been used successfully are not reused for subsequent authentications. For example, "publickey,publickey" requires successful authentication using two different public keys.
Note that each authentication method listed should also be explicitly enabled in the configuration.
The available authentication methods are: "gssapi-with-mic", "hostbased", "keyboard-interactive", "none" (used for access to password-less accounts when PermitEmptyPasswords is enabled), "password" and "publickey".
PubkeyAuthentication
Specifies whether to try public key authentication. The argument to this keyword must be yes (the default), no, unbound or host-bound. The final two options enable public key authentication while respectively disabling or enabling the OpenSSH host-bound authentication protocol extension required for restricted ssh-agent(1) forwarding.
PasswordAuthentication
Specifies whether password authentication is allowed. The default is yes.
Other settings interact with or apply to these settings as well. See man 5 sshd_config for the complete list.
Restarting Services After Configuration Changes
You will likely need to reload your SSH daemon or restart any muxed client connections in order for changes to configuration files to take effect. How you do this will vary from system to system, so please check your server OS and SSH client documentation for details.
