A Mary Sue who exists to be the perfect mate for a specific character. Far beyond Magical Girlfriend or Yamato Nadeshiko (which present extraordinary mates, but are played more-or-less realistically), this character has everything in the plot conspiring to enforce this One True Pairing. The author will generally either use this character as a stand-in for them to vicariously live out a relationship with the designated other half, or as their ideal significant other while they use a more "normal" character as their stand-in (sometimes both).
It's worth noting that since Wish Fulfillment places having a healthy, idealistic romantic relationship as one of its cornerstones, most Mary Sue characters will include an element of romance. They don't really become a Relationship Sue unless it becomes obvious that character exists, first and foremost, to be in a relationship with another character.
In Fanfiction, they are the perfect beloved of a canon character. Never mind if that canon character is already half of an Official Couple; this person is the real intended. The canon character who is to be paired with a Relationship Sue will usually be attracted to them at first sight; how long it takes for them to act on it depends on whether the writer is aiming for a W.A.F.F. or something angstier.
Gaining the canon character's attention (as they or the author will often profess) are qualities that the Relationship Sue almost never has. She could kill her beloved's pet kitten and mail it back to them, and they'd still tell everyone how in love they are with her for her modesty. Just as often, the author will develop the budding romance as a love that needs no reason, it just is. The worst situation for a canon character, however, is when at any point of the story they are made to suffer Character Derailment to become a more perfect match for a Relationship Sue. She will, in turn, almost always become a Clingy Jealous Girl when she eventually gets to be the character's girlfriend, but it is played as ideal.
If the canon character is already paired, or has another plausible-looking match in canon, this rival will either end up saying "I Want My Beloved to Be Happy" or turn evil; either way, he/she will probably undergo severe Character Derailment. That, or a nearby train will undergo severe derailment and splatter the character's canon love interest. Guess who will be there to comfort them in their time of grief. Alternately, if the writer can get away with it (or sometimes even if they can't) the canon match won't even be mentioned, or the would-be rival suddenly falls in love with somebody else (often turning gay in the process to make sure that it can't be undone). If they're already in an established relationship, expect the canon love interest to cheat on the desired character.
Anyone in the story who isn't a possible romantic rival, if they like the character whom the Relationship Sue is having the relationship with, will like him/her, too—regardless of whether they normally would. They may even be attracted to him/her—but they will not stand between him/her and his/her beloved if they intend to remain heroic/good characters. If the Sue likes them enough, she might even pair these characters up with other canon characters or even some OCs.
This can overlap with all of the other Mary Sue types, but the most common is good old fashioned Purity Sue. Fixer Sues can only fix up other pairs if they're this sort, unless they know they will get removed from the storyline (and stay that way). This type of character is always a Satellite Love Interest.
See also Strangled by the Red String, which often results in fans viewing a character this way.
No examples, please. This only defines the term.