As previously mentioned, IPv6 addresses are 128 bits long. This number of bits generates very high decimal numbers with up to 39 digits:
Such numbers are not really addresses that can be memorized. Also the IPv6 address schema is bitwise orientated (just like IPv4, but that's not often recognized). Therefore a better notation of such big numbers is hexadecimal. In hexadecimal, 4 bits (also known as "nibble") are represented by a digit or character from 0-9 and a-f (10-15). This format reduces the length of the IPv6 address to 32 characters.
This representation is still not very convenient (possible mix-up or loss of single hexadecimal digits), so the designers of IPv6 chose a hexadecimal format with a colon as separator after each block of 16 bits. In addition, the leading "0x" (a signifier for hexadecimal values used in programming languages) is removed:
A usable address (see address types later) is e.g.:
For simplifications, leading zeros of each 16 bit block can be omitted:
2001:0db8:0100:f101:0210:a4ff:fee3:9566 -> ¬ 2001:db8:100:f101:210:a4ff:fee3:9566
One sequence of 16 bit blocks containing only zeroes can be replaced with "::". But not more than one at a time, otherwise it is no longer a unique representation.
2001:0db8:100:f101:0:0:0:1 -> 2001:db8:100:f101::1
The biggest reduction is seen by the IPv6 localhost address:
0000:0000:0000:0000:0000:0000:0000:0001 -> ::1
There is also a so-called compact (base85 coded) representation defined RFC 1924 / A Compact Representation of IPv6 Addresses (published on 1. April 1996), never seen in the wild, probably an April fool's joke, but here is an example:
# ipv6calc --addr_to_base85 2001:0db8:0100:f101:0210:a4ff:fee3:9566 9R}vSQZ1W=9A_Q74Lz&R