It has multiple meanings, as it sounds like "A Door" and "Adore", each of which are relevant to the meaning (an opening that can be open or closed, and lavishing attention on something)
Somewhere between a friends with benefits and an open relationship with a boyfriend/girlfriend, a.D.O.R. is someone you have a romantic relationship with frequently but not exclusively. The "Distance" can be either as a result of a desire for independence or geographic distance that allows for coupling outside the relationship but doesn't weaken the passion of your interludes with that one person, similar to polyamory.
Differs from open relationships in that partners are not technically dating and isn't a friend with benefits because you are more passionate/romantic than is typical of that definition.
Al: "I do, we're a.D.O.R. We do what we want, and when we come together, it's amazing."