Most non-national Top-Level Domains (TLD) are called Generic TLDs (or gTLD). These do not have geographical or national restrictions.
The 'core' gTLDs are .com, .info, .net and .org: these are available to everyone. Other gTLDs like .biz, .name and .pro are restricted gTLDs because they have restrictions over who can register them.
The third group of gTLDs are sponsored by organisations: originally, these were mainly government ones like .edu, .gov, .int, .mil, but more were gradually added, and this year many new gTLDs like .london and .photography have been released.
The registry for an individual gTLD varies: the some core ones like .COM and .NET are administered by Verisign (under contract from ICANN), but others are directly administered by IANA (.int for example, for international organisations like NATO) and sponsored gTLDs will have a variety of registries.