Yes, it seems "Public" is the inbound, and private is your computer. The IP Address is simple enough, it's just the network address of the computer.
I've never actually encountered any port forwarding where the inbound and private ports were different numbers, but the same rules should apply. It seems the only difference is that the Apple router has a different look, and a different name for "Inbound Ports".
edit: Forgot to add the service list bit. The answer is No, you don't have to choose one. Those seem to be pre-made ones. For example, if you select E-mail or something, it may automatically fill in TCP port 21. If it doesn't let you save without choosing one, then choose something, then edit the numbers and save.
From my understanding "Inbound" would open a port on the router itself, since it's the main filter for the internet. From there, the "private port" would be the port it tries to send the information to on the computer or device. If that port happens to be blocked by the computer's firewall, then it won't work.
The private IP is just the network IP that the router uses to tell which computer is yours. I recommend setting up Static IP for doing things like Port Forwarding.