From the context in which I read the words, I think it refers to "your partner," said by a woman. What feelings are associated with S.O.? Camaraderie? Love? Affection? Friendship? Desire?
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Significant Other. It is anyone with whom you have a romantic relationship.
S.O. can be used inclusively for those with more common relationships and those with less common relationships. It would cover straight, gay, multigendered partners, legally adulterous partners during marital separation or could even be used outside of humanity (a dog party could involve bringing the owner's dog and the dog's S.O.).
I have never heard it used plural (in the case of bigomist), but it probably could be.
It does not mean a close friend with whom you do not have a romantic relationship. So a member of a "bromance" (two very close men, but not romantically involved) would not qualify as an S.O.