you have two ip addresses
wan - (wide area network) your router to the internet connection - 1 address shared by everyone connected to your router
lan - (local area network) your router to your machines
1 address per machine
DNS connects the two together so that you get the information you ask for - and others get only their stuff
wan - unless you are paying a premium for a static (wan) external address - it's dynamic - (which makes hosting an internet site impossible without a dyn dns service)
lan
most routers expect dynamic by default so are set up with DHCP to allocate the LAN addresses
in both cases DHCP allocates addresses for a set period (lease)
windows also defaults to dynamic
so see
control panel | Network Connections - right click your network adapter | properties
in the general tab - scroll down/click on internet protocol tcp/ip | properties
see what it says - obtain automatically - dynamic
or if you see numbers - static
and yes - your ISP's records show which address you had when - so it can be traced (but generally only the police would do)
finally - each network device has a MAC address which (in theory at least) is unique (sort of a fingerprint)
it tends to be mac addresses that are used in prosecutions
because it IDs the machine and not just the site
phew
obviously this is a potted version with more holes than my weekday undies - hopefully it covers your question