Online:A Friend In Need

Quick Walkthrough
# Talk to Canthion at the bank in Vulkhel Guard (Auridon), Daggerfall (Glenumbra), Davon's Watch (Stonefalls), or Vivec City (Vvardenfell).
# Find Szugogroth in the city's Outlaws Refuge.
# Find Bulag Idolus in Naril Nagaia (Greenshade), Orc's Finger Ruins (Rivenspire), Chid-Moska Ruins (Shadowfen), or Pulk (Vvardenfell).
# Return to Canthion.
# Craft, purchase, or otherwise acquire a piece of furnishing for your home.
# Meet Canthion outside your new home in Vulkhel Guard (Auridon), Daggerfall (Glenumbra), Ebonheart (Stonefalls), or Vivec City (Vvardenfell).

Detailed Walkthrough
There are a number of ways to acquire this quest:
Collect the free    from the Crown Store;
Find the Housing Brochure located near most Banks;
Visit Canthion directly at the bank in Vulkhel Guard (Auridon), Daggerfall (Glenumbra), Davon's Watch (Stonefalls), or Vivec City (Vvardenfell).

Locate Canthion and talk to him about the house for sale. Unfortunately, it turns out his previous tenant, Bulag Idolus has absconded with the key and the deed, after Canthion found out she was running a skooma operation from the property. Szugogroth, her associate, may know where she is. Look for him in the nearby Outlaws Refuge.

Szugogroth is initially unwilling to cooperate. You will have one or more options available to convince him otherwise:
Bribe him with a leveled amount of Gold.
Tell him you are in the Thieves Guild. 
Tell him you are in the Dark Brotherhood. 

He will eventually say that he searched her place the night she left, and allow you to look through the things he found. They are in his bag, where you will find a map and a scrawled note.
:'''Szugogroth's Bag:  

These will direct you to Naril Nagaia (Greenshade), Orc's Finger Ruins (Rivenspire), Chid-Moska Ruins (Shadowfen), or Pulk (Vvardenfell).

Head there and delve into the depths of the dungeon, fighting through the occupying bandits on the way. Eventually you will come across Bulag Idolus, who surrenders immediately, and complains about the protection that she paid for from the recently-deceased bandits. You then have to decide whether to kill her for the deed, or say you will let her go if she gives it to you.

One way or the other, you now have the key and the deed to Canthion's property, and can return to him. When you arrive, he tells you to keep the key and the deed, and meet him at the property. On the way, he asks you to pick up some Furnishings for your new place. You can either visit one of the furnishers, find a plan and craft your own, or visit a Guild Trader to see what others are selling.

Once you have acquired a piece of furniture, meet Canthion outside Mara's Kiss Inn Room (Auridon), The Rosy Lion inn room (Glenumbra), The Ebony Flask Inn Room (Stonefalls), or Saint Delyn Penthouse (Vvardenfell), and talk to him to complete the transaction.

The property is now yours, and you receive the Anthology of Abodes Available for Acquisition, which lists many other Player Houses that can be bought across Tamriel.

Notes
Completing this quest is required in order to purchase most other homes, so even if you do not particularly want the inn rooms (all of which are too small to do very much with), you should do the quest anyhow in order to enable you to purchase something more spacious.  If nothing else, the inn rooms (like all housing) represent a free teleport from anywhere in the world without needing to exit a dungeon or find a wayshrine.
If you want to acquire Saint Delyn Penthouse, you must begin the quest in Vivec City.  Otherwise, you will get the room that corresponds to your Alliance.  You can only acquire one free inn room per character, but if you have one character in each Alliance and a fourth in any Alliance for Morrowind, you can acquire all four rooms for free.  And since like all homes they are shared across all characters in your account, you will then be able to access all of them with every character.  Alternatively, you can purchase the other ones for   a piece.
This quest was removed in Update 16 and replaced with Room to Spare. This quest is similar but just requires you to talk to Felande Demarie and then, after a brief talk, she gives you the inn.

Bugs
 

 
Quest Stages