The global address system for IP Internet protocol) has over 4 000 million unique addresses divided in blocks of different sizes, so called subnet's. A subnet can content anything from 1 address to thousands of addresses. By combining the subnet's to larger blocks every operator can announce just one address (if continuous) and the table of nets hold by the Internet hardware (routers) can be held to a minimum.This is also the reason why the operators "owns" address blocks from which they provide their customers with addresses. You can see your IP-address on this link. You can also see to who which operator has registered this address on the address authority in different continents. The lookup can be found at respective regional Internet registries, see below.

IP can not by it's own send messages or data over a physical media (cable or air) but need a link protocol in between, a commonly used protocol is Ethernet but also HDSL and ADSL is used over cupper wires.

During the nineties there was a worry that all the addresses on the Internet should be used up, there was only a limited number to use and those could not be used in an optimal way. Simultaneity there was a stronger need to protect the local networks and more firewall functions were developed and used. By a combination of address space that were not allowed to use on the Internet (private addresses) and address translation in the firewall's (only one address used on the Internet) the demand for public addresses reduced dramatically and the address crisis was over for the time being.

Nevertheless the development of the next generation IP (IP version 6) that was to be the solution of the address shortness continue so when there will be an address shortage again there is a solution. However the commercial part of the Internet are not willing to pay for the migration today as there are no immediate needs.

A number of addresses in this system is reserved to be private and not to being used on the Internet. This way you can use not registered addresses on your local net whiteout rising that they are used on a host you need to reach (you can't have the same source and destination address). Private addresses must always be translated to public, registered addresses before traversing the Internet, something your firewall perform without you notice with NAT (Network address translation).

The global responsibility for assigning global addresses is owned by the organization IANA. It is delegated to regional authorities around the world. Here is the links to the different authorities:

Europe (RIPE)    North America (ARIN)    South America (LACNIC)    Asia (APNIC)    Africa (Afrinic)

If you want more detailed information about the IP addressing 3com has wrote a very good article.
The normal procedure to assign a single address to a computer is done automatically with a system called DHCP (Dynamic host configuration protocol). Read here about DHCP in Windows server.