In a /r/askreddit thread someone cited this as being a social unacceptable thing to do. During my time in the Navy I was told this was the reason for it..."It originally started from when sailors where eating on ships and they would put their elbows on the table to keep their dish from sliding during rougher seas. As this became a habit for them they would also do it while not out to sea and at pubs and such. Well sailors didn't have the best reputation for being clean, and well-mannered so non-sailors didn't want to be associated with them and thus would not put their elbow on tables. Hence where this rule came from!"
Is there any truth to this, or is it likely made up?
The "rule" goes back at least two millennia to the Wisdom of Sirach. While the mods want an in-depth and comprehensive answer to this question, it doesn't really raise one. It seems to boil down to this: while eating is necessary, how we eat is a custom or social ritual. In the case of elbows on the table, it appears guarded. Think of placing your elbows to either side of your plate and shoving meat into your mouth while looking furtively around the table.
Later examples of this behavior is said to have come from sailors who would use their elbows to stabilize their plates in rough seas, but typically anyone who felt the need to guard their food while eating (i.e., poor or lower-class folk while not at home, people who lived on the street, or who spent time with violent/aggressive people such as prisoners) would similarly use their arms to guard their food while they ate. In fact, my friend's brother went to prison and when he came out, it took him a long time to stop sitting like that. So, because of the association with fear/aggression/the lower-class, this behavior became considered rude at a polite dining table. In general, for a polite dining experience, we want to try to shed any of our behaviors that relate to being guarded and instead emphasize behavior that gives the appearance of comfort, relaxation, and openness.
Victorian teachings also noted that this behavior fell into other similarly aggressive behaviors such as dangling an arm over your neighbors chair or leaning too far to one side or the other while sitting: these all sharing in common the idea that you are taking up too much of other people's space (by leaning, dangling, or setting your elbows up like you're about to go MMA on your dining companions).
All that being said, put your elbows on the table if you'd like. In most cases, unless you're striking a very defensive pose with said elbows, no one is apt to notice. The main rule for this sort of etiquette is simply to make sure that you comport yourself in such a way as to not detract from the enjoyment of the meal for your co-diners. If you lightly rest your elbows on the table while trying to listen to someone, etc., and you're not spreading them wide and threatening all hands that come near you with a stabby-fork, you're probably going to be just fine.
edit: just added clarification to the end of the 2nd para. edit2: I a word.