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Background
You'll receive this quest from Soram at the
Dryad Outpost (#1).
Walkthrough
Soram will tell you that his wife Eliza went out to pick berries the other day, but that she hasn't been seen since. He'll speculate that she might have visited one of the sanctuaries to the south, but he won't know for sure, and he'll ask you to find her.
You'll find Eliza in the
Blasted Valley (#4), camped out in a ruined traveler's sanctuary protected by a magically-sealed door (#2) and a guarded door (#3) -- meaning that you won't be able to reach her until after you've completed the primary quest
The Mage's Tutor. However, Eliza will have become bound during her absence, and she'll attack you on sight. Eliza will drop
Eliza's Locket (Quest Item 7) when she dies.
When you return to Soram, he won't exactly be thrilled by your report, and he'll attack you. After you've beat on him for a bit, he'll give up the fight, but he won't forgive you, and that's how the quest will end.