IP address
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Summary
An IP address is a unique value you are given when connecting to a net using the Internet Protocol.
IPv4
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.
IPv6
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:
34fe:3f25:8d34:0000:46d9:0903:47ef:32aa
When all four hex digits in a group are 0s (0000), that group can be omitted or replaced by a single 0. For example:
34fe:3f25:8d34::46d9:0903:47ef:32aa 34fe:3f25:8d34:0:46d9:0903:47ef:32aa
Leading 0s can be omitted. For example:
34fe:3f25:8d34::46d9:903:47ef:32aa
If more than one consecutive group has only digits with value 0, they may all be replaced by only two colons. For example:
34fe:3f25:8d34:0000:0000:0000:0000:32aa
would become
34fe:3f25:8d34::32aa
IPv4 addresses can be easily written in IPv6 by doing ::ffff:IPv4-address. For example:
::ffff:192.168.0.1
Notes
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
More information about it can be found at Wikipedia's IP article and Wikipedia's TCP/IP article.