[Bad Karma[CORE]]

Static IP
This IP never changes, hence the "static" part. Some ISPs assign static IPs to their customers which has advantages and disadvantages. A disadvantage is if your IP gets blacklisted on some server. Then you can't access it anymore for the fact that even resetting the router/modem will not assign a new IP. An advantage is that you can conveniently operate a server from home (though most people don't because the upstream is way too low to run a server efficiently) or host websites not having to deal with DynDNS updates (hey...more tech speak).

IP Bans
IP bans work by blocking a single IP but it's not down to blocking single IPs only. You can block IP ranges, subnets and entire countries. With regards to the "banned on a forum" part -> it's actually quite easy to have banned users not signing up with a different IP. without going into too much details but for example on my site banned users receive what we call a "token of death" which basically is a cookie ban on top of the IP ban.