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An IP address is a unique value you are given when connecting to a net using the Internet Protocol.
Internally, IP addresses are 4 bytes numbers. Anyway, their common use is splitting this addresses into 1 byte numbers with a dot (.) between them. Since the range of values a 1 byte number can be is 0 - 255, the range of IP addresses goes from 0.0.0.0 to 255.255.255.255 which makes up to 2^(4*8) possible IP addresses.
This is the new revised IP protocol version, which is meant to replace the old limited IPv4.
It is mostly compatible with IPv4, either directly or indirectly (through middle-way translators).
IPv6 addresses are 16 bytes long and are normally written as eight groups of four hexadecimal digits, each group separated from the others by a colon (:). For example:
When all four hex digits in a group are 0s (0000), that group can be omitted or replaced by a single 0. For example:
Leading 0s can be omitted. For example:
If more than one consecutive group has only digits with value 0, they may all be replaced by only two colons. For example:
IPv4 addresses can be easily written in IPv6 by doing ::ffff:IPv4-address. For example:
Don't confuse this number with the eD2k IDs. IP addresses are to identify you anywhere in the net you are connected to, while eD2k ID's are only used to identify you on that eD2k server you are connected to.
You might also want to check what ports are.
More information about it can be found at Wikipedia's IP article and Wikipedia's TCP/IP article.