A loyalty mission would be like, having a warrior in your army/party/team and he/she just learned that the little brother/sister is sick with something.
the mission could be just to see the kid before they die, added effect if you have a mage with you, they know whats wrong and need a single herb to cure the kid.
if you just visit the kid and they die, you get +10? points to greater loyalty.
help save kid you get +25? points to greater loyalty.
the mission doesnt actually help the player much in games like Mass Effect or Dragon Age, its a personal thing for X character, but it can add depth to the characters and also improve their personal relationships with the player.
in the same games, if you do the loyalty mission, anyone that can be romanced by the player, will become fully available.
my thoughts is that just one mission feels a little cheep, that having 5 or more, would result in true loyalty and friendship, maybe more if you play your cards right.
think of a game like the sims where you have a friendship bar, if there was a second bar that indicates loyalty, the characters could not be friends with you but still loyal, in that they didnt like what you did, but understand you have your reasons kind of mentality.
I always liked the idea that romance, depending on the world you are in, should not be so typical, that there are other kinds of them of every spectrum possible with all the pros and cons within reasons.