This quest is one that comes after " The White Phial ", so go see that one first before thinking of doing this one. Once you complete that quest, a courier will come find you after a number of days to start this one.
Return to The White Phial alchemy shop in Windhelm and tell Quintus that you got his letter. He will confirm that he thinks he can repair the phial, but he will need three ingredients: Mammoth Tusk Powder, a Briar Heart, and some Unmelting Snow.
o Mammoth Tusk Powder:
This ingredient is only known to be produced by giants. Yikes. Well, hopefully you aren't doing this quest TOO early in your adventure, as giants will just decimate a low level character. The marker will send you to Stonehill Bluff, a giant camp north of Whiterun and south of Dawnstar. Head over there and you can either kill the giants to ensure some leisurely searching on your part or you can run in, find the powder in a stone basin in the middle of the camp, nab it, and run out. Your choice.
o Briar Heart:
Honestly, I already had one of these when this quest started, so I didn't have to go find one. I have to imagine that the quest will send you to a Forsworn Camp if you don't have one, in which case you're looking at killing 6-7 forsworn and making sure to loot the briarheart that you find. Keep your health up and bring a companion for distraction if need be and you'll get this ingredient easily.
o Unmelting Snow:
For this ingredient you need to head to the Throat of the World (which means that you must have access to it and have completed the Main Quest SOMEWHAT (past "Elder Knowledge"). Once you have (and you DO need to...), head to the Throat of the World and follow the marker to a snowbank that you can active to get the snow. An easy ingredient.
With all three ingredients in hand, return to Quintus and let him know of your success. He will take the ingredients and repair the phial, then head upstairs (follow him) and show his success to Nurelion. Nurelion will be happy, but then he will die (did NOT see that coming), leaving just you and Quintus. Quintus will say he has no desire to keep the phial and will give it to you, but due to his lack of skill he is only able to make it refill with ONE type of liquid and will ask you what you want it filled with. Below is a list of what you can choose and what your choice means, which is handy since you only find out AFTER you choose what you want:
**- I'd like it to have the power of healing. (Restore 100 points of Health.)
I want to resist the forces of magic. (Resist 20% of magic for 60 seconds.)
I want to be tougher in battle. (Stamina is increased by 20 points for 300 seconds.)
I want to strengthen my magical skills. (Magicka is increased by 20 points for 300 seconds.)
I want to deal more damage in battle. (One-handed weapons do 50% more damage for 60 seconds.)
I want to be better hidden in the shadows. (You are 20% harder to detect for 60 seconds.)**
And, like we've been talking about, the White Phial will refill once a day with a message popping up in the upper left of the screen to let you know. After picking what potion you want, the quest will be over. Enjoy your bottom-less potion!