The domain name system had been run by a small group of individuals and organisations under US government contracts. In 1998, ICANN was created as a non-profit to coordinate it. Within a year, the system opened up so that many registrars could compete for registrations, and prices fell.
This is the reason you can today choose among hundreds of registrars, and why transferring a domain is a standard process rather than a negotiation.
The change looks administrative, and in a sense it was. But the structure it created is the one every domain owner still deals with, usually without knowing it.
Who looked after names before 1998
The DNS itself was designed in the 1980s. Its coordination was handled informally by researchers, with Jon Postel at the University of Southern California best known for running the function called IANA, which kept the lists of protocol numbers and the top of the naming tree. Registration of the popular generic names, .com, .net and .org, was done by Network Solutions under an agreement with the US National Science Foundation. By the middle of the decade that arrangement was creaking. Demand for names was growing quickly, the company was the only place to buy them, and people increasingly disagreed about who should be in charge of something so central.
The 1998 transition
In June 1998 the US government published a policy paper proposing that management of names move to a private, non-profit organisation with international participation. ICANN, the Internet Corporation for Assigned Names and Numbers, was incorporated in California in September 1998, and in the months after it signed an agreement with the US Department of Commerce to take on the work. Jon Postel died in October 1998, shortly after the new body was formed.
ICANN did not run the whole internet, and it still does not. Its remit was the narrow, important parts that need one answer worldwide: the root of the naming system, the allocation of address blocks, and the contracts with the companies that operate top-level domains. In 2016 the US government's oversight role formally ended.
Opening the market
The big practical change came in 1999. The monopoly on .com was split into two jobs. One was the registry, the master database that says which names exist and where their name servers are. The other was the registrar, the company that deals with customers. A shared system let several registrars write to the same registry, and ICANN set up an accreditation process for them. A first, small group began operating in 1999, and many more followed.
Prices dropped, quickly and by a large margin, and the service around the name improved. Registrars began bundling DNS hosting, email forwarding and web hosting. Hosting companies became registrars, or resold someone else's service, which is why your hosting provider may also be where your domain lives.
Rules that came with it
Competition needed ground rules, and the ones set around this time still apply. Accredited registrars follow a common transfer policy. Trademark conflicts go to a dispute procedure, the UDRP, adopted in 1999, which is faster and cheaper than court. The public record of who registered a name (WHOIS) became a standard obligation, though it has since been restricted by privacy law and gradually replaced by a newer protocol.
The result for an ordinary owner is a short list of steps, the same wherever you bought the name.