n addition to the global top-level domains, there are a whole bunch of two-level domains for the different countries of the world. Some of them let people register directly under them as second-level domains, while others have a more deeply nested structure. Some limit registrations to residents of the appropriate country, while others are open worldwide. Much other variation in domain policy exists.
.us -- US Country Domain
The United States of America's .us domain is registered via nic.us, and has traditionally been subdivided geographically. Other than a few specialized categories such as community colleges and Native American tribes, most users were supposed to register under their hometown, with domains like yourname.shreveport.la.us, so that what you register is a fourth-level domain under your city (or county), state, and country. These are much cheaper to register than the .com domains (often free, though this varies by locality), but are lengthier. However, these domains might be a good choice if you want your site to be geographically identified. Many people register domains like joeschmoe-of-milwaukee.com where joeschmoe.milwaukee.wi.us would have been more logical. The usual objection is that more users are familiar with .com domains, but that's self-fulfilling; of course people won't be familiar with .us domains if nobody ever uses them, but if lots of people start using them as site addresses, public awareness would follow, and people would become better-educated about the hierarchical structure of the domain name system in general.
However, that structure has now changed -- a new registry provider has been selected by contract of the U.S. Department of Commerce (the winning registry is the same one that runs the new .biz TLD), and they are now offering registrations directly at the second level, like yourname.us. They had a trademark-owners-only sunrise phase (similar to that of the .info TLD, but only for owners of U.S. trademarks, not globally), followed by first-come, first-served registration, for which various registrars took queued requests over the weeks and months preceding that. Names like help.us -- with.us have some "cute" subdomain possibilities, like grow.with.us or fool.around.with.us. But don't try anything ending in "-r.us... the Toys "R" Us people have a very aggressive legal department.
Legislation that recently passed Congress will establish a .kids.us subdomain in which only child-friendly sites can register. No act of Congress, however, was needed for the registrant of protozoa.us to offer subdomains for sites that are safe for protozoans.
Neustar, the company that runs the .us domain, has recently been placed in charge of China's .cn domain as well, and will be throwing that domain open to registrations without any Chinese nexus requirement.
Similarly to .us, Canada had a hierarchical system for its .ca domain, but allowed companies and organizations that are nationwide in scope to register directly under .ca, and province-wide organizations to register within a province subdomain (like .on.ca), instead of making everyone register within a city. In 2000, a new plan liberalized this to allow unlimited registration directly under .ca (as is now being done in .us).
Both .us and .ca require some sort of connection with their respective countries in order to register there. Individuals wishing to register must be either citizens or residents; multinational corporations need to at least have a contact address in the given country.
Countries other than the United States and Canada rarely used geographical hierarchies; they usually, right off the bat, allowed either direct second-level registration or registered at the third level beneath logical subdivisions like .co.uk and .org.uk for commercial and noncommercial sites in the U.K. respectively. They vary a lot as to the degree, if any, of local presence needed for registration by foreigners.
It's not actually necessary for an entity to be a real country in order to have a country code. Antarctica has the .aq domain. Another not-really-a-country with a country code is the Palestinian Authority, with .ps (they used to have a site at gov.ps, but it hasn't worked lately, maybe because of all the problems they've been having at their headquarters these days). The European Union is about to inaugurate a .eu domain. There's a big debate going on about whether to terminate the still-existent .su domain for the Soviet Union.
Other Country Codes
Some country-code domains are open to registrations from people having no connection with that country, and this can be used as a revenue source by small countries with little need for the namespace domestically. Tonga (.to) is one such domain that's available to people worldwide. Turkmenistan made its .tm domain available internationally, with particular appeal to companies with trademarks ("tm"), but registrations have been put on hold after the Turkmenistan government objected to some "obscene" domains being registered. More recently, Tuvalu's .tv domain was sold (actually leased for a 10-year period) to an American company for millions of dollars, and they're now auctioning off the most desirable names there rather than just letting anyone grab them cheaply like other registrars generally have done. There are similar registries at .fm and .am, belonging to the Federated States of Micronesia and Armenia respectively.
.tk, belonging to the island nation of Tokelau, is giving away "free" domains (supposedly in keeping with the tribal values of that island where everything is shared), but there are lots of catches attached, and many of the "good names" are reserved to be registered at a price.
Since some of the country codes are two-letter words in English or other languages, this has been taken advantage of by various people who have set up "redirection services" under addresses like go.to, here.is, i.am, start.at, or for Spanish-speaking people, pagina.de and espacio.de. The idea is to follow such an address with your site name: http://go.to/mysite, which is set up to redirect to your real site address. Many of these addresses are available free, but you're at the mercy of the service that provides the redirect, which could go out of business or impose a really annoying popup ad on visitors.
The television networks and other things that get .tv domains could be in trouble in a few years -- the island of Tuvalu is in the process of sinking into the ocean due to global warming, and its population already has plans to evacuate. If the country stops existing, then that will open the debate about whether its domain should be deleted, just as that debate is now progressing over the .su (Soviet Union) domain. Thus, .tv registrants should probably keep up on what happens to .su as an advance warning about what might happen to them.
DNS Server Requirements
The global TLDs (.com, etc.) require two DNS server addresses in the registration record (a primary and a secondary), but the registrars don't actually check that those servers respond to requests for the domain; you can get away with registering domains with anybody's server in the record, though they won't actually work unless whoever's in charge of that server sets up the DNS over there. Country code domains can vary in their requirements. The Mexican domain registrar, for instance (.mx), requires only one DNS server address, but actually checks it during the registration process to make sure it responds to requests for that domain, so you need to get your hosting provider to set up DNS before you register the domain. This may be one of the reasons there is less cybersquatting and other domain abuse in the .mx domain than in the global ones (but then again it may just be that cybersquatters and domain abusers, even when they're Mexican, don't care as much about .mx addresses as .com ones...)
As with other domain names, there have been some conflicts over country code domains.
One interesting conflict occurred in 2000 as ICANN got set to launch new top level domains. As one of the proposals (ultimately accepted) was for a .biz domain (for businesses), a company in St. Louis sued ICANN, claiming to have obtained contract rights from the country of Belize to operate its .bz country code registry, and that a .biz domain would be confusingly similar to it and hence a violation of trademark law. ICANN disagreed strongly with this claim, saying that .bz was for the use of the country of Belize, not to stand for "business", that country code domain registries were delegated the responsibility of running such registries, but didn't own the domains involved, that the U.S. Patent and Trademark Office has specifically denied the trademarkability of top level domains, and that the complaining company wasn't even the current delegated operator of the Belize TLD. The court refused to issue an injunction to stop the .biz domain.
There's also an ongoing debate about ICANN taking control of the country code domains, most of which existed long before ICANN was around, and many of which resent having a US-based organization be able to dictate rules and demand payments from them. In 2005, the .au country code domain (Australia) was taken away from its long-standing operator, an individual associated with a university, and given to an Australian nonprofit group founded to operate that domain. Since then, the new organization has agreed to sign an agreement with ICANN, and ICANN has a forum section to discuss it.
The expansion of the DNS to character sets outside the "US-ASCII" letters and numbers is a hot topic these days. This isn't technically related to country code domains (while these new "internationalized" domains may be used in country codes, they're also being used in global TLDs such as .com), but it does relate to making the domain name system more international in nature.
A company called I-DNS claims to be making available domain registration in foreign languages using character sets that aren't legal for traditional domain names. They do this by using servers that translate such "foreign domains" to ASCII character strings. And the resulting translated domains, like L6FDP645L316L7DFL40D.L16CL3F8, seem to have top-level domains like .L16CL3F8 (an ASCII encoding of a Chinese string that presumably means something similar to .com), which don't actually exist in the "official" domain name system. This is still an experimental system which might catch on in countries where it's useful, but could fragment the domain name system if it does. Registrations for Chinese domains under this system are being taken at chinese-dns.com. There's already a heated political conflict, as the Chinese government wants monopoly rights to domains in the Chinese language worldwide. That's not really fair; they have control, properly, of the .cn country-code domain, but shouldn't have any specific control of any other domain, even if it's in Chinese, any more than England should have control of anything worldwide that's in the English language.
There are other, incompatible schemes in progress for internationalized domain names, some of which only work on particular browsers, and thus aren't really a full part of the Internet. There is a forum area on the ICANN site to discuss this topic. One current scheme for internationalized domains uses the RealNames resolution system, which presently only works for users of Microsoft Internet Explorer, leaving other browser users out of luck and also preventing use for such things as e-mail and FTP.