Is my ip adress my computer or my internet connection?

[Who]

if you connect to the internet via a router
the router allocates you an IP address which only applies within your local area network (ie, devices connected to the router)
the router has its own IP address which your ISP allocates it each time it sets up a connection to the ISP (normally) - so this could be different each time it connects

The router IP is the only address that shows to the rest of the internet

Similarly if you connect a second computer then the router will allocate it its own IP address only applicable in your local area network.
the router IP address will not change.

The function of the router is to keep track of all computers that want to connect to the network, and "route" all input/output to the correct computer.

if your computer connects to a modem then to the internet. then your computer is allocated its own IP address by your ISP (if its dynamic then it could change each time you connect)