The IP address is given to you by your internet provider... It also comes with a name.. something pretty cryptic usually like NW32544.or.bell.com. The IP address actually changes once in a while, but the name won't.
Most software (forums, etc) will log both.
Here's the catch.. the name is linked to the physical address.. So if you get banned while working from your Ipad or computer at home, that will be what gets banned... You take your Ipad to work, and you can probably get back on (using a different email, username, etc), but if someone is on their game and you log in from your home, they can see that and ban you again (this time also banning your work IP)... If you're really bad, they can ban your entire area.