So a few days ago I came up with the idea of hosting my own web server for my files. I've installed everything on my computer, but, while looking for my IP address, I noticed something strange: all the sites for checking IP had different answers. After some testing, I noticed that when I connect my PC directly to my (Motorola SB501) modem, there is only one IP told by all the sites, but when connecting to my (Tenda 11N) wifi router, the issue persists. Therefore, I believe this is not an issue with DHCP/Static IP, as the IP is static when connected to the modem, but I believe there is something with the router or its configs. And by the way, resetting the router to factory settings did not help!
Your modem router has two ip addresses. One is your internal connection to the router and can normally be set by you. This is known as the LAN address (Local Area Network).
The other is the address by which the outside world sees you (the WAN - Wide Area Network) and is dictated by your Internet Service Provider (ISP). Unless you have specific arrangements in place, the ISP address is normally dynamic, meaning every time you connect to the Internet, your ISP will allocate an address from as pool of addresses that it has. That is probably why your address is changing, as it's pot luck what is allocated to you.
You can still set up a server in a mickey mouse sort of way by using dynamic addressing in your router. But it wouldn't really be a practical solution for a busy web site.