Magic is real and widely believed in the world of Song of Ice and Fire. Yet it is also rare enough for many to doubt its existence or at least its effectiveness, particularly in Westeros and most of all in the Citadel.
Sorcery is a sword without a hilt. There is no safe way to grasp it.
Magic is a little understood force in the world. It has been so long since magic was truly potent that most understanding of it only lives on in superstition and rituals of questionable validity.
In Westeros and most of the lands west of Valyria, magic is held to be a mythical force only. The Maesters of the Citadel may apply for a Valyrian Steel link for the study of magic, but they believe it to have faded away following the Doom of Valyria, since there has been no record of its use in Westeros since that time. However, recently the magical Valyrian Glass candle in the citadel has ignited anew and it seems that the potency of magic in Westeros has increased, and miracles and abilities that were long thought lost now appear to be returning, although this is commonly neither known nor accepted.
Blood, life, and death seem to be keys that can help unlock the secrets of magic. Warlocks from Qarth bathe Sam Tarly in aurochs’ blood to make him brave. Mirri Maz Duur sacrifices the horse, and so much more, in her attempt to restore Khal Drogo. Dany burns the maegi on Drogo's funeral pyre and wakes dragons.
Additionally, pyromancers claim that recently spells required in the creation of Wildfire have begun to work better and more efficiently than they have in many years, and dragon sightings are becoming more and more common in Essos  However, even before that in the lands east of Valyria magic has remained a potent force. In Qarth, the enigmatic warlocks of the House of the Undying, said to posses great powers and in the lands by the Jade Sea and Asshai by the Shadow, where all manner of sorcerers, necromancers, spellsingers, aeromancers, and more are said to gather and practice their arts.
Some of the abilities exhibited by those who practice magic are:
- Elemental control: The priests of R'hllor seem able to control the element of fire in order to engulf their weapons in flames, or to use the fire alone as a means to attack enemies.. Also in Qarth Daenerys saw a firemage creating a ladder of flames that stood up to forty feet high. It is possible that other wizards and sorcerers know ways of using other elements.
- Divination: Some people exhibit the ability to see the future either through looking in the fire in the case of the followers and priests of the Lord of Light or through dreams.In the second case this is called Greensight if following the way of the Old Gods or dragon dreams if it is done by a Targaryen.
- Skinchanging: The ability to enter the mind of an animal and control its actions.It is a rare ability with only one man in a thousand being born a skinchanger. Having the blood of the first Men usually makes it more likely for someone to be a skinchanger.
- Raising the dead: Some priests of R'hllor have been capable of performing the extraordinary feat of raising a man from the dead. However the reborn person may have difficulty remembering parts of his past life such as Beric Dondarrion or they exhibit a change in personality like Catelyn Tully. What red priests generally fail to do is to heal a man brought back to life by whatever injuries he may have suffered and as a result their bodies are battered by their wounds.
Other than the Children of the Forest, who said to be strong in magic, native magic in Westeros is much rarer. In days past, the greenseers of the children possessed incredible powers over nature; Skinchanging, slipping into the minds of beasts, and Greensight, the power to foretell the future. Though the children of the forest and their greenseers have long passed from Westeros, there are still descendants of the First Men who possess those powers and while those powers are not something you can will to occur, it seems that those who hold the old gods are more susceptible to becoming one.
All Valyrian magic was rooted in blood and fire. It is said that they could set dragonglass candles to burning with strange, unpleasantly-bright light. With the obsidian candles, they could see across vast distances, look into a man's mind, and speak with one another though they were half the world apart. It is often said that the old wizards of Valyria did not cut and chisel stone, but worked it with fire and magic as one might work clay. The people of Valyria were very strong in magic, and they would use their powerful wizards and their dragons in conjunction with their armies to conquer most of the eastern continent. Dragons were controlled by whips, magic horns and sorcery.
Bloodmages from the Shadow Lands beyond Asshai are said to use the darkest of all magics, and possibly the most powerful. They have dark spells to save a man from death, though some say death is cleaner.
There are firemages who can conjure ladders from the air that are made of fire and can stand forty feet high. The mage can climb up the ladder, each rung disappearing behind him, leaving nothing but silvery smoke. When he reaches the top, the ladder and he are both gone. 
References and Notes
- A Feast for Crows, Prologue.
- A Storm of Swords, Chapter 76, Jon XI.
- A Feast for Crows, Chapter 45, Samwell V.
- A Storm of Swords, Chapter 54, Davos V.
- A Game of Thrones, Chapter 23, Daenerys III.
- A Feast for Crows, Chapter 36, Cersei VIII.
- A Game of Thrones, Chapter 64, Daenerys VIII.
- A Clash of Kings, Chapter 40, Daenerys III, page 426.