'The relationship' refers to one's partner, usually female but not exclusively, whether it be girlfriend, fiancée, wife or their male counter-parts. As long as said partner is more than a passing fling then they qualify as 'the relationship'.
To be used as an alternative to your partners name as well as my/the missus, partner, wife, girlfriend, etc.
Sam: "So who's coming around for drinks?"
John: "Jimmy, Sandi, Graeme, The relationship, Will and Laura."
Sam: "That's a cool shirt. Where did you get it from?"
John: "Not sure. The relationship got it for me."