As someone who has some pets this is basically what I can explain from it.
First of all it is not necessarily sexual and in my case it is entirely a non-sexual thing, some people may take it there but that's not the thing WE mean when we say owner or pet in the forums sense.
An owner pet relationship mainly falls under the concept of submission and caretaking, the level of commitment or seriousness varies as Ventus said, but the concept more or less stays the same. In my case the relationship was something I took seriously, but was also very casual, I saw it like a close friendship.
The basic thing is the person who is the pet is generally charged with being cute and cheering up the owner, the owner takes a protective stance of the pet, helps the pet with his or her problems and just generally acts as a sort of mentor and protector. In exchange the pet has loyalty to the owner, and is charged with doing what you'd expect a pet to do, cheer you up, be loyal and understanding and what-not. That's how I took the pet-owner relationship, but anyone will take it different. A parallel to this is a family, some people will call their facebook friends or online friends "brothers", to signify the closeness of their relationships, a pet and owner relationship is similar but has that level of caretaking and submissiveness in it, similar to a father and son, but not quite THAT far.
It's a state of kinship, I would say above a friend but below a relationship, a pet and owner must have trust with one another and the relationship can get very intimate and affectionate.
It's a difficult thing to describe, but I hope that makes some sense.
