Immortality is the ability to live an eternal life and to never die. Some magical creatures are born immortal. There are number of ways to achieve immortality.
Cornelius Sigan's jewel[]
Merlin emerges victorious out of his confrontation with Sigan.
This jewel allowed the very powerful sorcerer Cornelius Sigan to achieve immortality. He presumably cast a spell to transfer his soul inside the jewel and to contain it. If someone touched the jewel, Cornelius Sigan's soul would have possessed him. Anyway, this way to achieve immortality didn't allow Sigan to keep or preserve his body and his soul could be trapped in the jewel once more with a powerful spell (The Curse of Cornelius Sigan).
Cup of Life []
Sir Leon fails to kill one of Cenred's men, due to him being immortal.
When a person's blood is put in the Cup of Life, the person can become immortal. They cannot be harmed by ordinary weapons and possibly by magic as well. However, when a person toys with the Cup's very powerful magic, he becomes a living dead, lacking independent thought and will, and can be killed by a sword forged in the dragon's breath, like Excalibur. The only other way to kill this person is to empty the Cup of his blood, for the person is no longer bound to the Cup's power (The Coming of Arthur).
Necromancy[]
One of the immortal skeletons summoned by Morgana.
Another way to create the magical state of immortality is by using necromancy. This is one of the forms of darkest magic and can be used to conjure the dead. Having already passed away, they cannot die, but this may not count as true immortality as the subject is externally animated rather than being fully independent.
Portal[]
A portal allows a magical creature to travel between the world of the living and the Spirit world. The manticore was a creature which did this. When in the living world the manticore's life force was bound to the Spirit world, possibly meaning that it was immortal but it could not remain in the living world for a long time and had to return from where it came. Also the manticore could be killed if the portal was destroyed by magic.
Sidhe[]
The Sidhe are immortal creatures and they could live for many years. However they were not fully immortal as they could be killed by powerful forms of magic and could be cursed to live a mortal life. This form of immortality only extended their lifespans.
Morgana[]
It was never mentioned or shown that Morgana was immortal, however, she couldn't be killed except by magic. However, in The Sword in the Stone, a Camelot soldier stabbed her with a normal sword, and almost died but Aithusa healed her. With as exceptionally powerful as she was, she probably could've used magic to live longer.
Merlin's magic[]
Merlin's magic was so powerful he was able to achieve immortality as shown in The Diamond of the Day as he was capable of living forever as he was still alive even in modern times. His immortality may be because of his powerful magic or his destiny that ties him with the return of Arthur. Merlin's appearance had changed in the final scene and he appears to be an old man, but whether he is permanently old or if he is able to revert himself to his former youth is unknown.
The Triple Goddess[]
Being "God" of the Merlin universe, she is eternal.