Remote connection with a public IP

You have to add a port-forward rule in your router and set the destination IP address of the required computers LAN IP and port 22.

You can not specify to SSH which computer you want to connect to, you can only connect to an IP address (or a resolvable domain name) and a port (default 22). Only to those for which you set a port-forward rule on your router.

However, you can add multiple rules to each computer but the WAN side port must be different for each one, ie. you can use port 22 only once.

For example:

• WAN port 22 ---> LAN IP: 192.168.1.1 PORT: 22
• WAN port 2220 ---> LAN IP: 192.168.1.2 PORT: 22
• WAN port 2221 ---> LAN IP: 192.168.1.3 PORT: 22

See my answer to a very similar question: Why can't I access my web server from outside the network?

A router has multiple IPs. Its public IP is, of course, public. That is the address that you give to any website you connect to so the server knows where to send a reply. A router also has a gateway IP, which is typically the address on your LAN which other devices behind the router see. This IP is private in that it does not exist on the public internet, but it is not secret (most people use the same range).

My question is, is the router's IP really public?

It is public, but you don't have to worry about it. Knowledge of an IP address is typically not a problem unless you need anonymity. You don't have to worry about being hacked or tracked down based on IP. I explained in another answer why there's no need to worry about having a public IP address. If the address wasn't public, then you wouldn't be able to hear a response from any site you connect to!

Or is there another level of NAT in my ISP's network?

Some ISPs have another level of NAT called Carrier-Grade NAT, or CGN. If your public address as seen by the router (not by websites that check your IP) is in the range of 100.64.0.0/10 (see RFC 6598), then you are most definitely using CGN. If you have any doubts, call your ISP and ask them.