ICANN is an international private-public partnership founded in 1998. ICANN is a non-profit. They manage the internet’s Domain Name System (DNS), including both generic (gTLD) and country code (ccTLD) Top-Level Domain names and the numeric address infrastructure. There are seven original gTLDs (.com, .edu, .gov, .int, .mil, .net, .org) and in 2001 and 2002 more were added including four unsponsored (.biz, .info, .name, .pro) and three sponsored (.aero, .coop, .museum).
Sponsored sTLDs are limited to narrow communities for which the sponsor sets the rules generally. Since then several more were added (.asia, .cat, .jobs, .mobi, .travel). Each TLD has its own registry maintained by the registry operator.
As a user you will buy your domain name from a registrar who then send the information to the registry.