A numerical label assigned to a device on a network, used to route traffic and identify endpoints. Every server has one (or more). Every request your browser makes includes the server's IP address in the response headers—meaning your ISP, hosting provider, and anyone else on the path knows where the traffic goes.
This matters for domain registration because the IP is where your DNS records actually point. If you're hosting a site, your registrar's DNS servers translate your domain name into the IP address of your server. If that IP is static and tied to your name, you're discoverable. If it's shared (virtual hosting), you're one of thousands on the same box. If it's offshore or bulletproof, it may tolerate abuse complaints better—though "better" is relative.
Reverse DNS (PTR records) goes the other way: given an IP, what domain does it resolve to? Law enforcement uses this. So do network operators checking mail server legitimacy. If you're running a mail server, the reverse IP should match your domain or it gets flagged as spam.
The TLD registrar you choose doesn't control your IP, but it controls the DNS records that point to it. No DMCA compliance from us means we won't yank your domain because someone complained about the IP's content—that's your hosting provider's problem, not ours.