Nearly every device that is connected to the Internet has an IP address. There are a few devices which function as passthrough that do not have an IP address, but 99% of Internet users will never come in contact with one of those devices, so let's pretend that they do not exist for the moment.
Therefore, your computer has an IP address which uniquely identifies it within some particular scope. In this case the scope is your local network....maybe a WiFi connection to your house, or an ethernet connection in your office, etc. In any case your IP address associated with your computer has a local scope. Your local network is connected to the Internet using a router. A router is a network device used to connect one network to another. The router has its own IP address. In addition to the router, there will be a modem (note, many ISPs provide devices which bundle the router and modem and often a switch into the same device, but each device type must be present regardless of whether you have 1 box sitting beside your computer or three). The modem which is assigned an IP address by your ISP is your local network's interface to your ISPs network. You may have heard the term, Wide Area Network or WAN. From your standpoint, the LAN is your local network, and the WAN is anything on the other side of the device provided by your ISP.
Remember, I said your computer's IP address has local scope. In other words, only things on your local network can see your computer's IP address. The modem provided by your ISP has a scope of within the ISP's network. But, the ISP uses one of those network devices which does not have its own IP address to deliver an external IP address with global scope to your ISP provided modem. This means that your modem's external IP address truly is in the global scope.
Websites like WhatIsMyIPAddress.com can only "see" network devices with global scope. In other words, they can only "see" your ISP's external IP address associated with your connection. And that's what they report back to you. So you see a different IP address than the one assigned to your computer within your network.
Clear as mud?