IP address
Is how your computer is identified on the network. It's almost like a house and ZIP address for a house - it's unique on any given network
If you are connected directly to the internet (for instance), it will be a public IP which others can reach, such as 123.456.78.90
If you are behind a router, the router will take over the public IP (123.456.78.90) and assign your Mac a private IP, for instance 10.0.1.2, which is visible only to the computers on your private network created by the router, but to nobody outside on the internet.
TCP port
Is a Transmission Control Protocol. It is a protocol which unifies how computers talk to the outside world, to say it bluntly. It is like a door to / from your computer.
So, you computer has an IP address which is almost like a house address and a zip and a country code - it's unique to every PC on the network.
Your TCP ports are like a bunch of doors going in and out of your house - in order to use them, they must be open. Each door serves a different purpose.
There is a door #80 for internet browsing, there is a door #22 to make secure connections to other computers, there is a door #21 for sending files via FTP, there is a door #25 for your e-mails, door #5298 for your instant messaging, etc....
The same address and door logic applies to anything on the net, including your router. So, if you have one, you have to make sure it has the right ports open and that there isn't something what keeps opening or closing them on a schedule or randomly.