IPv6 Unicast Addressing
RFC4291 details the IPv6 addressing architecture. In terms of unicast addresses, the addresses you really need to be aware of are.
In IPv4 this is 127.0.0.1 of course. In IPv6 the loopback address is simply
Link Local Addresses
Similar to IPv4's APIPA addresses, a link local address is scoped soley to your local L2 segment. The most obvious difference compared to IPv4 though is that Link Local Addresses are mandatory in IPv6. (So even if you statically assign IPv6 addresses to your hosts, they will keep their Link Local Addresses too.)
Link local addresses have addresses in the range
Global Unicast Addresses
Essentially, this is the public IPv6 address space and is numbered
You can of course get Provider Allocated (PA) IPv6 space as you could with IPv4, but an option that is now available to a wider range of organisations, is to go to your regional internet registry and get your own Provider Independant (PI) IPv6 space.
Unique Local Addresses
RFC4193 defines Unique Local Addresses (ULAs), the IPv6 equivalent of IPv4's private addresses. Currently, ULAs should be allocated from the range
Although ULA's aren't routable across the internet, the recommendation is to randomly choose your prefixes from the range to try and keep the addresses unique globally. (Thus reducing the liklihood of needing to renumber the network in a merger scenario) There is a ULA prefix generator here to this end.
Whether or not you should use ULAs should be carefully considered. In IPv4, the default design in most cases, is to hide private address ranges behind behind edge NAT, however, this is borne out of necessity rather than desirability in most cases.
Ivan Pepelnjak has written a useful blog that discusses if/when you might need/want to use ULAs. (To be fair he's written useful blogs about pretty much every aspect of networking you could imagine)
IPv6 Basics Part 1 - Address Format & Types
IPv6 Basics Part 2 - Unicast Addressing
IPv6 Basics Part 3 - Protocol Differences
IPv6 Basics Part 4 - Multicast
IPv6 Basics Part 5 - Planning an IPv6 Network