Magic or sorcery in Camelot is an mysterious force linked to the old religion preceding Christianity in England.
Magic in Camelot is linked to the powers of nature, we see this when the knights are hunting and Morgan's shapeshifting weakens with each animals death.
Magic is a primal and visceral force, whilst Morgan relishes the power she does have to learn to take care as she weakened by the this power. Merlin who according to the story and confirmed by the writer is long lived has learned to not use this power but perhaps as an opposite extreme.
Thought it takes a vast amount of both inner and outer strength magic could be used for the benefit of others, but is always part of nature and therefore a part of it is unpredictable. A good example of this is when Merlin freezes a lake to walk across the water, but the process continues and leads to the death of Excalibur.
Magical Powers and Spells
Control the elements: Merlin states that he can to some degree control the elements of nature. We see this when he freezes an entire lake and makes a fire intensify.
Shapeshifting: The power to alter ones shape and form or that of another.
Visionary Magic: Merlin exhibits this power in various forms, dreaming the future and powers of foresight. The power to get visions from others by touching them.
Spell to see through the eyes of another: Morgan mixes some of Arthur's blood in water which she incants over and heats over a flame. Drinking this she is able to see through Arthur's eyes.
Spell of healing: It is alluded too that Merlin has powerful healing abilities, he uses this on Igraine to try and heal her, though she is close to death. It seems his own life force is being transferred when he attempts this, so Igraine stops him.
Scrying: Morgan, Sybil and Viviane work together over a bowl of water conjuring a vision of Igraine.
Impact of Magic
It seems the practice of magic has an impact on the sorcerers body, mind and soul. Merlin seems very aware of this. He suffered this after the incidents of 'The Lady of the Lake'
Morgan who suffers bleeds and other symptoms but in the episode 'The Lady of the Lake' we see Morgan nearly die in her over use of magic. Her mentor Sybil arriving in time to help her through it.
Humans who practice magic are often referred to as "sorcerers". Contrary to Christian belief sorcerers do not receive their powers from demons, nor do they worship the devil. It is unclear in the show how many sorcerers have a natural gift or how many had to study to attain their powers, it seems likely a balance between the two.
Merlin being extremely old is from a time pre-dating Christianity where Sorcery and the Old Religion held sway and primal powers were appeased by its Priests/Priestesses and sorcerers.
When we meet Sybil she is a nun who maintains the old Religion along with her Christian faith. Morgan clearly has strong connection to these powers and intends to bring this back to Britain in time.