Skyrim Dawnguard Side Quest Walkthrough: Dawnguard Rescue Missions

Talk to Florentius (you must have completed the quest Lost Relic to get the Rescue Quest). Ask what you can do to help. He tells you Arkay has grave news. Your wife/friend has been kidnapped. A vicious night stalker rules over a group of necromancers - seeks to gain leverage over the righteous. Arkay's never clear on life and death. Is she still alive? There's always hope...

This starts the quest:

Rescue:
Florentius Baenius has learned that a master vampire has kidnapped (my wife/husband/friend) and is keeping her prisoner in Fellglow Keep.

Rescue (Wife/Friend) From Fellglow Keep (Or other random location)

The victim of the kidnapping is chosen at random, to be either your wife/husband (only if you are married, of course) or some character that has a friendly disposition towards the dragonborn.

Go to the dungeon you were assigned. Fellglow Keep is Northeast of Whiterun, but you could have gotten any other dungeon across Skyrim. Fight your way to the end of the dungeon. Kill the Master Vampire, then you need to rescue the spouse/friend, who is usually locked in a holding cell nearby.  If you have a good sneak and/or range attack, you might want to kill the enemies near the hostage Stealthily and Quickly, to avoid risking the prisoner (who is your spouse or friend) getting killed by them in the heat of combat.

When you are done, return to Florentius, and tell him you have rescued the person. He gives you a random reward. You may get additional rescue missions from him, sometimes even you might need to rescue a person whom you've already rescued, but got kidnapped again.

This completes the quest, Rescue.