How is my IP address assigned?

Summary: Every device on an TCP/IP network must have a unique IP address. IP addresses are assigned, either automatically by DHCP, or by manual configuration.

Given that IP address must be unique for every computer attached to the internet,how is it possible that we do not plug in IP addresses for the computer we want to connect?

•

Well, every device connected to the internet must have a unique IP, it's true. And they're assigned one of two ways: static or dynamic. But there's also a useful trick that lets multiple computers share a single IP address ... and that trick is called a router.

•

Static IP addresses are exactly that: static or unchanging. They are assigned by your network administrator or ISP, and yes, you do have to configure the computer or other internet device manually to respond to that specific address.
But, as you point out, most folks don't need to do that. So how do they get their IP addresses?
Enter the dynamic IP address and "DHCP" or Dynamic Host Configuration Protocol.
Using DHCP (which is the default for Windows TCP/IP connections) the computer broadcasts a special request for an IP address to the network. An upstream device, commonly belonging to your ISP, responds with an IP address that the computer then configures itself to use. Especially when many computers aren't connected continuously, this allows the ISP to reuse the IP addresses of computers that have disconnected from the internet.
If you need your computer to be identifiable on the internet ... for example if you're running a web server or want people to be able to connect to your machine, you'll probably need or use a static IP address. On the other hand, if all you do is connect out to surf or read email, as most users do, then a dynamic IP address is the easiest to configure.
Routers are devices that allow multiple computers to "share" a single IP address. The device that's connected to the internet is the router, and it has a unique IP address. The router can then act as the DHCP server to the local network handing out local IP addresses to the computers connected to it. As traffic flows across the router, it does the job of translating the IP addresses from the local addresses it has assigned, to the external IP address it was assigned, and routing the right bits of data to the right computer ... hence the name.
One of the many side effects of using a router is that it can be assigned a static address on the internet, and hand out dynamic IP addresses locally, vice versa, or any combination.