Option B - Bring the Star to Anarea
Note! If you prefer to give the Daedric artifact to Nelacar, skip to Option A - Bring the Star to Nelacar.
Open the world map and return to the Shrine of Azura. Once again use the stairs to reach the altar, approach Anarea and hand her oven the broken Star (screen above).
New main objective: Speak to Azura
Let Anarea cleanse the artifact and soon afterwards you will have the chance to speak to Azura herself. Interact with the Star lying on the altar (screen above).
New main objective: Tell Azura you're ready to enter the Star
From the conversation with z Azura you will learn that the only way of repairing the Star is defeating Malyn Varen, who has placed his soul inside the artifact. Azura will offer to temporarily transfer your soul into the Star, so that you can fight Malyn on his own terrain. Note! Before agreeing to move your soul to the Star (screen above), make some preparation as for obvious reasons your teammate won't be able to help you in the new area. I'd suggest permanently increasing your fire and electricity resistance, or at least bring potions and spells that can give you a similar effect. Additionally it would be good to have powers, shouts or spells that let you summon creatures to help. It's also worth to save your game, as you will be able to able to back out of the Star in case the battles in it prove too hard. Later on you can return to the Altar and try again.
New main objective: Destroy Malyn Varen's soul
You will be transported inside the Azura's Star. Speak with Malyn Varen (screen above) and the mage will start running away. You will have to catch and defeat him.
DON'T try to catch up with him at once, as quickly eliminating him will be pretty much impossible. What's more, you could run into some big trouble as you will have to face one or a couple (up to three) Kynreeve Dremoras (screen above). They're quite a challenge, as not only do they have lots of hit points, but can also use powerful spells. The creatures should by default attack you with fireballs, but there's always a chance that they will use a different spell.
You need to fight the Dremoras very carefully, as otherwise you will get surrounded or get under massive fire and die quickly. You should make use of the mentioned summon, whichever type you have (screen above). It probably won't deal too much damage, but will draw their attention away enough for you to lead out attacks.
If you prefer close combat, it would be good to stick to one chosen Dremora and keep attacking it (screen above). The Unrelenting Force Shout might come in handy, as even though throwing them into the abyss is impossible, knocking them down to the ground will still make them easy targets.
If your character prefers ranged weapons, lead out precise attacks and keep constantly strafing to avoid getting hit. If you're more into magic, I'd suggest focusing on fire spells (screen above). Regardless of the chosen tactic, keep retreating to the starting point each time you kill a Dremora to get some rest and regenerate health, magic and stamina.
Head after Malyn Varen only after getting rid of all the Dremoras. Reaching the mage shouldn't take you much time (screen above).
You should begin the fight with the Unrelenting Force Shout (screen above) or any other attack that will knock him out or temporarily paralyse. Be ready to heal yourself often, as the mage will attack you with powerful lightning spells. Luckily however, he won't summon any creatures to help so you will be able to use your own for your advantage and numerical superiority.
Keep fighting until deprive Malyn Varen of his whole health bar. Azura (screen above) should now speak to you. Wait to get transported back to the world of the living.
Now just speak to Azura for the last time. The Daedric Lady will give you the Azura's Star (screen above), using which you will be able to use souls to enchant items. You will also be able to ask Aranea Irenith to join your party as a full-time follower.