It really is a lot more straightforward than it sounds. If you get a camera that is specific to home surveillance, it probably comes with instructions and apps to handle everything you need to do. I just used a webcam that I had on hand and kind of hacked it together. Here's some more detail about what I did.
The camera picks up an IP address when you plug it into the network. I logged into the router's web interface (usually 192.168.1.1) and looked at the list of connected devices. Found the camera in the list and noted it's IP (if you have software that comes with the camera, it can probably find it for you). Logged into the web interface of the camera and configured it to always take the same IP (static IP). Once I know where the camera will always be, I went back to the router and went into port forwarding (it's a little different on every router, but is always there if the router is worth anything). I set port 2222 from the WAN(outside IP for the modem -173.19.90.242 in my case) to forward to port 80 (default) on the IP address of the camera. That allows any request to the outside IP for my modem at port 2222 to forward to the camera on port 80 automatically. If you google your modem/router model and port forwarding you can probably find specific instructions for what you have, but that's essentially what you need to do if you configure it by hand. Like I said though, if you pick up a home surveillance camera package it probably has more automated ways of handling things.