Wights are dead men or creatures raised up by the Others, seemingly when touched by the cold that accompanies them.  Anyone who falls against the Others must be burned, or else the dead will rise again as their thralls. Fear of their own dead becoming wights leads the wildlings to burn them.
The appearance of wights depends entirely on the condition of the corpse when it is raised. Some are lifelike, while others are badly rotten although the process of decay has been halted. All are easily identified by their eyes having turned bright blue  like two blue stars  and their hands and feet black and swollen with pooled and congealed blood. 
Wights are supposedly attracted to warm blood and will attack with surprising strength. Unlike the Others they are slow and clumsy.  Wights have a queer, cold scent that can panic animals if they catch a wiff of it. 
It appears that wights retain at least some of their former memories, since the wight of Othor seemed to know where Lord Commander Jeor Mormont slept and who he was as it tried to kill him, and the wight of Thistle recognises Varamyr Sixskins even warged into the body of a wolf.
Being dead, wights feel no pain and will continue to fight regardless of injury.  Though they can be stopped by total dismemberment, their limbs will continue to move if detached from their bodies. When a wight is destroyed, the blue disappears from its eyes.  
Wights appear to be able to remain in a state of stasis, such as outside the hillside entrance to the cave of the three-eyed crow. The snows have buried most of the dead men, but they are still there, hidden, frozen, waiting. 
Dragonglass cannot stop a wight  but fire can. They are highly flammable and will be quickly consumed if set aflame.  It is unknown at this time whether wights can cross the Wall on their own, although corpses brought through the Wall can apparently still reanimate as wights. Burning corpses appears to prevent them from being reanimated by the Others to become wights.
The Seven-Pointed Star
A list of individual characters that have been turned into wights over the course of the series:
A Game of Thrones
A Clash of Kings
Othor’s hand (still moving) was sent to King's Landing in a jar with Alliser Thorne to try to an impress on the king the severity of the threat from beyond the Wall so that the king may send more troops to the wall. Tyrion Lannister kept Ser Allister waiting so long that Othor's hand decomposed, and was not very convincing.
A Storm of Swords
During the Battle of the Fist of the First Men the Night's Watch was unable to defend against the wights that were sent against them. After a crushing defeat, the majority of the Night's Watch who fought on the Fist were slain by the insurmountable odds.
A Dance with Dragons
While listening to gossip in the Lazy Eel in White Harbor Davos notices that no one speaks of King Stannis. No one seems to know he has gone north to help defend the Wall. Wildlings and wights and giants had been all the talk of Eastwatch, but no one seems to be giving them so much as a thought.
Jon consigns some of the dead wildling corpses in the ice cells.  Sometime thereafter he talks with Bowen Marsh, Septon Cellador and Othell Yarwyck in his chambers. Bowen Marsh eventually asks Jon why he has kept those corpses in the cells as they make the men uneasy. Bowen also asks why the corpses are kept under guard. He wonders if Jon fears they will rise up. Jon replies that he hopes they do. Septon Cellador is horrified, saying to Jon,
Wights are monstrous, unnatural creatures. Abominations before the eyes of the gods. You … you cannot mean to try to talk with them? 
Jon replies by asking septon Cellador if they can talk. Jon says he thinks not, but he does not claim to know. He tells the men,
Jon’s answer does not please the men.
Cotter Pyke sailed towards Hardhome by the command of Lord Commander Jon Snow. By rescuing the wildlings Jon hopes to be able to prevent the Others from raising an enormous army of wights by killing the thousands of wildlings stranded at Hardhome and raising them as their thralls. Jon knows that if the Others manage to do this the wights will head for the Wall in their hundreds and their thousands.
Later on Jon Snow goes down to the ice cells where they have four living men and two dead men in. Jon had almost forgotten the corpses. He thinks that he had hoped to learn something from the bodies they’d brought back from the weirwood grove, but the dead men had stubbornly remained dead.  When Jon Snow decides to move the living men he instructs Bowen Marsh to leave the corpses . Jon thinks to himself that he will no doubt need to burn them eventually but for the nonce they are bound with iron chains inside the cells, and that that, and being dead, should suffice to hold them harmless. 
Yet his eyes were still open. They stared up at the sky, blue as sapphires. 
The world shrank to two blue stars, a terrible crushing pain, and a cold so fierce that his tears froze over his eyes. 
References and Notes
- A Game of Thrones, Chapter 52, Jon VII.
- A Game of Thrones, Prologue.
- A Storm of Swords, Chapter 33, Samwell II.
- A Storm of Swords, Chapter 46, Samwell III.
- A Dance with Dragons, Chapter 4, Bran I.
- A Storm of Swords, Chapter 18, Samwell I.
- A Dance with Dragons, Chapter 34, Bran III.
- A Feast for Crows, Chapter 27, Jaime III.
- A Game of Thrones, Chapter 52, Jon
- A Dance with Dragons, Chapter 35, Jon VII.
- A Dance with Dragons, Chapter 39, Jon VIII.
- A Dance with Dragons, Chapter 58, Jon XII.
- A Dance with Dragons, Chapter 69, Jon XIII.
- A Game of Thrones, Chapter 53, Bran VI.
- A Game of Thrones, Chapter 52, Jon VII, p 555.
- A Dance with Dragons, Chapter 28, Jon VI.