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What is an IP address?

An IP address (Internet Protocol address) is a unique address that certain electronic devices use in order to identify and communicate with each other on a computer network utilizing the Internet Protocol standard (IP) - in simpler terms, a computer address. Any participating network device - including routers, switches, computers, infrastructure servers, printers, Internet fax machines, and some telephones - can have their own address that is unique within the scope of the specific network. Some IP addresses are intended to be unique within the scope of the global Internet, while others need to be unique only within the scope of an enterprise.

An IP address provides details of the internet provider that provides the internet access, it does not refer to an individual computer. For example, if you change your service provider for the internet - your IP address will change.

In other words, the IP address acts as a locator for one IP device to find another and interact with it. It is not intended, however, to act as an identifier that always uniquely identifies a particular device.

Each machine on the network must have its own, unique IP address. Similarly than each house has a unique mail address, if a network is connected by the rest of the world, that address must not only be unique within the local network, but unique within the rest of the world, also. By the most common IP version (IPv4), IP addresses are 32-bit values, which are represented usually by four sets of numbers, ranging from 0-255 separated by dots. This is referred by as dotted-decimal notation. By dotted-decimal notation, an address might look like this:

192.168.42.1

Because all these numbers range between 0-255, those can be represented by eight bits and are therefore referred by as an octet. This IP address is often thought being composed as a network portion (at the beginning) and a node (or machine) portion at the end.

IPv4 only uses 32-bit (4 bytes) addresses, which limits the address space to 4,294,967,296 possible unique addresses. As the number of addresses available is consumed, an IPv4 address shortage appears to be inevitable in the long run. This limitation has helped stimulate the push towards IPv6, which is the new standard protocol for the Internet. Addresses are 128 bits (16 bytes) wide, which, even with a generous assignment of netblocks, will more than suffice for the foreseeable future. Example:

3ffe:2f80:3912:1

Source: Wikipedia.
Further reading: IP addressing, RFC791.