There are many special relationships around the world, though. :) Both the US and the UK have relationships with other countries that have been described as "special" by some (the US with Israel, the UK with Canada, etc). It's basically just something to use in diplomacy and political science to emphasize the bonds (diplomatic, economic, cultural, etc.) between two nations. It's not an official agreement like the Entente Cordiale or other ententes, alliances and treaties of "We're Gonna Be Friends, This Time For Sure! (Until Something Happens)," there's no "hereby we declare our relationship Special" (except for Churchill, because, yeah). It's just something to throw around to emphasize how very good friends the two countries are.

Of course when it coems to the US and Britain it has a special (hur hur) meaning that is meant to describe their relationship, but basically it's really just two words that only have sentimental/diplomatic value.