There is no computer system directly connected to the internet. All traffic is done via routers, switches, gateways and (god forbid) hubs.
Inside a single network, every system has its own unique identification (ip adress and mac address). Outside that network they use one or more connections to connect to other networks, which do the same.
You can set up a system to automatically give each device it's on ip address (DHCP) or make a ip adress manually (static).
There are 13 root networks that define the core names of the internet.
What if I buy a new PC, and give away my old one to my 13 year old nephew? Does that mean the IP goes to his house?
Say he moves to Kentucky, IP remains same as in NYC?
So does my new PC have the same IP, or a new one?
Is my old PC (now in another state) assigned a new IP as 2 nodes cant have the same IP?
They're not in the same network
If the ISP assigns a new IP every time you log on, then how can the FBI track anyone?
It is always known which ip connected to which other ip on a certain time. It is also always known which device had which ip and MAC address at that time.
Does the ISP keep a database of every IP for every user ever logged on?
IPv6 may be set up differently, because there are enough ipv6 addresses to give every molecule on this planet a unique identifier. However, most ipv6 networks follow the same path as ipv4.