Reasons why Shiva Has a Snake Around his Neck

1. First Story - 

The Snake around lord Shiva’s neck is called Vasuki. Vasuki was the Serpents king.

Lord Brahma worries about Shiva 's annoyance, and its consequences on the universe when Shiva decided to not to interfere in universe affairs. Lord Vishnu knows this is because of separation of Shakti from Shiva.

Vasuki's role

Vasuki decides to take care of Sati(Shakti). Vasuki wishes to approach Lord Shiva for Sati's welfare. Vasuki visits Kailash to meet Shiva.

Vasuki informs Lord Shiva about Sati's suffering. He urges Shiva to take care of her.

Vasuki pays his gratitude to Shiva for blessing him.

Now on other side all the snakes were already given order by Lord Brahma that snakes and serpents will live only inside earth(underworld), they will not come on land surface. But because Lord Vishnu wanted to arrange meeting of Lord Shiva with Sati(Shakti born to Daksha Prajapati's daughter) , he caused an unbalance to world. Snakes came on the earth surface.

But this was against Lord Brahma's orders, so Lord Vishnu sent his vehicle Garuda, a bird, to kill all the snakes.

The episode begins with Garuda attacking all the snakes, which creates among all snakes.

There Vasuki tries to stop Garuda from attacking the snakes, but Garuda reminds Vasuki of the curse which Brahma has given to them, and also tells him that snakes have crossed their boundaries and that snakes should stay in Patal-loka. But Vasuki defends the snakes and garuda tells that he would kill Vasuki. And a fight starts between Garuda and Vasuki.

Garuda picks Vasuki and dashes him on the ground. Vasuki calls Shiva and says that you can’t tolerate injustice, then why so delay in coming as Garuda is killing Vasuki.

Shiva comes there to save Vasuki, which leaves Garud shocked.

Shiva decided to stop Garuda, Lord Vishnu appeared to save Garuda.

Lord Vishnu said "Please Lord, you have to win me first to kill garuda".

For the sake of Vasuki and Snakes ,Lord Shiva accepted Lord Vishnu's challenge. Lord Vishnu reveals to Lord Shiva that he had plotted to fight with him for the welfare of the universe. Lord Shiva and Lord Vishnu reconcile with each other. Lord Vishnu succeeds in pacifying Lord Shiva.

Vasuki's Sacrifice

Lord Shiva blesses Vasuki and Garud. He takes Vasuki around his neck because Vasuki's role and his sacrifice was too much important. Vasuki was injured by Garuda and he was devotee of Lord Shiva. Vasuki was the King of Serpents, and he lost many snakes because of this incident, many of family members,too.

Lord Shiva takes Vasuki around his neck and blesses him that he will be having more important role while "Samudra Manthan"  will be happening in near future.

2. Second Story - Protection of Snakes

Snakes are meant to be dangerous reptile and is feared by many of us. Lord Shiva show us that he controls fear and death by wearing snake as ornament.

It is also said that the snake three rounds around Lord Shiva's neck depicts future, present and past. That states that Lord Shiva is Independent of Time and controls time.

There is a story that when the snake species were in endangered, they approach Lord Shiva for Shelter. Lord Shiva gave them shelter by letting them stay in Mount Kailash. But due to cold weather, the snakes used to approach Lord Shiva for warmth of Body. Thus, He as a protector used to wore these snakes as ornament to provide warmth.

3. Third Story - Samudra Manthan

After the Samudra Manthan and before the immortalizing Amrita came out. There were a number of things that came along with it one of them being Halahala, a lethal poison. 

This terrified the gods and demons because the poison was so powerful that it could destroy all of creation. Then the gods approached Shiva for protection. 

Shiva, being the daring God that he is, consumed the poison in an act to protect the universe, and his wife Parvati grabbed Shiva's throat in an effort to prevent him from swallowing the poison, which was the most deadly poison in existence, harmful even to a god. 

As a result, Shiva's throat turned blue. For this reason, Lord Shiva is also called Neelakantha (the blue-throated one; "neela" = "blue", "kantha" = "throat" in Sanskrit). 

As it is not feasible even for a goddess to hold on to anyone's neck forever. It is believed that she tied the snake around his neck, so as to prevent him from swallowing the Poison.