Who is in charge of domain names?

Internet Assigned Numbers Authority (IANA) [Marina Del Rey]
http://www.iana.org/
The IANA's role is to allocate IP addresses from the pools of unallocated addresses to the RIRs according to their established needs.

The IANA webpage will send you to the other top-level domain authorities. E.g., it sends you here for .edu:
http://www.educause.edu/edudomain/

The ones for the country code domains (such as .uk or .fr) are a bit hidden:
http://www.iana.org/cctld/cctld-whois.htm
(Two letter domains, such as .uk, .de and .jp (for example), are called country code top level domains (ccTLDs) and correspond to a country, territory, or other geographic location.)

The Internet Corporation for Assigned Names and Numbers (ICANN)
http://www.internic.net/
http://www.icann.org/registrars/accreditation.htm
Domain names ending with .biz, .com, .info, .name, .net or .org can be registered through many different companies (known as "registrars") that compete with one another.

Only registrars accredited by the Internet Corporation for Assigned Names and Numbers (ICANN) are authorized to register .biz, .com, .info, .name, .net and .org names.

ICANN is the new non-profit corporation that is assuming responsibility from the U.S. Government for coordinating certain Internet technical functions, including the management of Internet domain name system. More information about ICANN can be found at
http://www.icann.org